Voraussage des Verteilungskoeffizienten in der Flüssig-Chromatographie mittels COSMO-RS

Transcription

1 Voraussage des Vertelungskoeffzenten n der Flüssg-Chromatographe mttels COSMO-RS - Predcton of the partton coeffcent n lqud chromatography usng COSMO-RS Der Technschen Fakultät der Unverstät Erlangen-Nürnberg zur Erlangung des Grades D O K T O R I N G E N I E U R vorgelegt von Martn Rethnger Erlangen

2 Als Dssertaton genehmgt von Der Technschen Fakultät der Unverstät Erlangen-Nürnberg Tag der Enrechung: 16. Aprl 2012 Tag der Promoton: 26. November 2012 Dekann: Prof. Dr.-Ing. Maron Merklen Berchterstatter: Prof. Dr.-Ing. Wolfgang Arlt, Prof. Dr.-Ing. Andreas Fröba

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5 Zusammenfassung De vorlegende Arbet befasst sch mt der Vorhersage des Vertelungskoeffzenten verschedener Eluenten nnerhalb flüssg-chromatographscher Trennsysteme. Zel st es, enen Ansatz zur Modellerung solch enes Trennsystems zu entwckeln, welcher also m Stande st das Molekülvertelungsverhalten zwschen ener flüssgen moblen und ener komplexen statonären Phase vorherzusagen. Bsherge Vorhersagemethoden we de Quanttatve Structure Retenton Relatonshps (QSRR) baseren auf ener Velzahl anpassbarer Parameter welche de physkalsch chemschen Egenschaften der moblen sowe der statonären Phase des betrachteten Trennsystems beschreben. Auf Grund der dadurch notwendgen emprschen Parameteranpassung haben QSRR Methoden enen nur begrenzt prädktven Charakter. Das Conductor-lke Screenng Model for Real Solvents (COSMO- RS) ersetzt dese anpassbaren Parameter durch ene auf Quantencheme und statstsche Thermodynamk beruhende Herangehenswese und brgt somt de Möglchket ener ren molekülstrukturbaserten Vorhersage der thermodynamschen Egenschaften sämtlcher Systemkomponenten. Zuerst wurde der Enfluss verschedener Molekülkonformere auf COSMO-RS- Rechenergebnsse untersucht. Herbe wurden expermentelle Daten des Oktanol-Wasser Systems herangezogen und es zegte sch, dass enzelne Molekülkonformere sgnfkanten Enfluss auf das Rechenresultat und somt auch auf de Vorhersagequaltät haben. In enem weteren Schrtt wurde de Lestungsfähgket der COSMO-RS Vorhersage m Hnblck auf de Anwendung bezüglch flüssgchromatographscher Trennsysteme mt Umkehrphasen untersucht. Zu desem Zweck wurde de komplexe statonäre Phase als pseudo-flüssg angenommen und somt de Möglchket eröffnet, dese mt Hlfe von so genannten pseudo-flüssgen Molekülen zu beschreben. De Betrachtung der statonären Phase als en Pseudoflud brgt somt den Kerngedanken des Projektes. Der auf Molekülstruktur baserende Ausgangspunkt aller COSMO-RS Rechnungen machte es grundsätzlch möglch, alle denkbaren pseudo-flüssgen Molekühlstrukturen zu entwerfen. Verschedene Screenng-Versuche zegten: Um de Wechselwrkungscharakterstka ener Umkehrphase nachzuempfnden st es en snnvoller Ansatz, wenn man en pseudo-flüssges Molekül generert, welches verschedenen Fragmenten der realen statonären Phase nachempfunden st. Neben dem Vorhersageansatz mttels pseudo-flüssger Moleküle wurde untersucht, n wewet es mt dem COSMO Modell möglch st, ene QSRR Methode zu entwckeln. Herzu können von COSMO genererte Deskrptoren, so genannte σ-momente, für jede berechnete

6 Molekülstruktur abgeletet werden. Vergleche mt expermentellen Daten zegen, dass σ - Moment baserte QSRR ähnlche Vorhersagequaltät errechen, we der Ansatz mt pseudoflüssgen Molekülen. Bede m Rahmen deser Arbet beschrebenen Methoden, de Vorhersage des Vertelungskoeffzenten mttels pseudo-flüssger Moleküle sowe de σ- Momente baserte QSRR wurden n deser Art zum ersten mal auf de Vorhersage des Trennverhaltens flüssgchromatographscher Systeme angewandt.

7 Summary Ths thess examnes the predcton of partton coeffcents of dfferent elutes wthn lqud chromatographc separaton systems. The man goal s the development of a phase modellng approach that s capable to predct molecule dstrbuton between a bulk moble and a complex reversed statonary phase. Prevous predcton methods such as Quanttatve Structure Retenton Relatonshps (QSRR) are based on several adjustable parameters used to descrbe physco-chemcal propertes of the moble and the statonary phase n the separaton system examned. Due to the emprcal parameter fttng, QSRR methods have lmted qualtes n terms of true predctvty. The Conductor-lke Screenng Model for Real Solvents (COSMO-RS) approach replaces adjustable parameters by a quantum chemstry and statstcal thermodynamcs based approach and allows for a merely molecule structure based thermodynamc property predcton of all components wthn the separaton system under nvestgaton. For a start, the nfluence of molecule conformatons onto COSMO-RS calculaton results was nvestgated. Therefore, expermental data of the octanol-water system was used and t has become obvous that sngle molecule conformatons have sgnfcant nfluence on calculaton outcomes and hence on qualty of predcton. As a next step, the ablty of the COSMO-RS predcton approach for modellng a two phase reversed phase lqud chromatographc system was nvestgated. For ths purpose, the complex reversed statonary phase was assumed as pseudo lqud and therefore modeled by so-called "pseudo-lqud" molecules. The consderaton of the statonary phase to be a pseudolqud, bears the central dea of ths thess. The nherent advantages of COSMO-RS led to the possblty of creatng any pseudo-lqud molecule structure magnable. Dfferent screenng experments have revealed that depcton of real statonary phase surface fragments (bound lgand plus part of slca surface) s a good approach to smulate the characterstcs of statonary phase nteracton. Addtonal to the pseudo-lqud molecule based predcton approach, t was nvestgated to whch extend COSMO s capable of developng a QSRR method. For ths purpose, COSMO generated descrptors, so-called σ-moments can be deduced from any generated molecule structure. Comparson to expermental data shows that σ-moment based QSRR wll reach predcton qualtes that are comparable to that of the pseudo-lqud molecule approach. Both methods descrbed wthn the work at hand, the pseudo-lqud molecule approach as well as

8 the σ-moments base approach were, for the frst tme, appled onto the separaton behavour predcton of lqud chromatographc systems.

9 Table of contents Table of contents 0 Enletung Goal of work Bascs Bascs of chromatography Chromatographc separaton prncple Volume and porosty n a chromatographc column Retenton tme and related quanttes Peak wdth and related quanttes The moble phase The statonary Phase The reversed statonary phase The reversed statonary phase: Technques of examnaton Bascs of phase equlbra Thermodynamc equlbrum Elute dstrbuton equlbrum n a chromatographc system The adsorpton equlbrum Descrbng adsorpton equlbrum usng adsorpton sotherms Lqud-lqud (absorpton) equlbrum Underlyng dstrbuton mechansm: adsorpton or partton Absorpton as underlyng molecular separaton prncple Modelng of chromatography Mass balance based models of chromatography Actvty coeffcents models Exothermodynamc models Chemcal and quantum-chemcal bascs Ab-nto and sem emprcal methods Densty functonal theory Contnuum solvaton models The Conductor-lke Screenng Model (COSMO-RS) COSMO calculated σ-moments Conformatonal analyss Models of statonary phases n RP-HPLC Expermental... 57

10 Table of contents 3.1 HPLC materals, equpment and expermental methods Solvents (moble phase) Solutes and tracer components Statonary phases Chromatographc system setup Expermental HPLC measurng methods k -factor determnaton from chromatographc measurements Lterature research Computatonal modellng: programs and computatonal methods Molecule geometry generaton and conformatonal analyss Converson of HyperChem output data DFT geometry optmzaton Calculaton of the chromatographc parttonng coeffcent usng COSMO- RS Conformaton selecton for COSMO-RS calculatons Selecton of conformatons wth equal dfference n E COSMO Results & Dscusson Conformaton selecton of solute and solvent molecules: Influence on calculaton results / Selecton rules Effect of conformaton selecton on γ Effect of solute and solvent conformaton selecton on K OW Dervaton of a conformaton selecton rule for solute and solvent molecules Development of reversed statonary phase modellng Development of pseudo-lqud molecules Effect of pseudo-lqud molecule structure and composton on predcton qualty Effect of pseudo-lqud molecule conformaton selecton on predcton qualty Effect of actve groups on predcton qualty Expanson of the lnear dependency between log k exp und log K COSMO-RS Applcaton of a pseudo-lqud molecule on dfferent statonary C18 phases Predcton of the separaton factor Extenson of statonary phase modellng onto normal phases Log k predcton va QSPR method by usng σ-moments Overvew of retenton predcton models and predcton methods

11 Table of contents 5 Résumé Reference lst Appendx A-1 Modellng results A-2 Program code

12 v Lst of symbols Lst of symbols Latn symbols Symbol Meanng SI-Unt A molecular surface area Å 2 A peak area A area m 2 a actvty a contact area Å 2 a senstvty b slope b Langmur parameter m³ k -1 c concentraton mol l -1 E electrc feld strength V m -1 E energy functonal J mol -1 E energy J e area related energy J mol -1 m -2 F degree of freedom F volume phase rato f fugacty Pa G Gbbs energy J g molar Gbbs energy J mol -1 g partal molar Gbbs energy J mol -1 H Hamlton Operator J H enthalpy J h molar enthalpy J mol -1 K partton coeffcent (based on mole rato) k capacty factor l number of adsorbed layers M molar mass g mol -1 m mass kg N plate number n amount of substance mol n number P pressure bar P partton coeffcent (based on concentraton rato)

13 v Lst of symbols p frequency Q reduced surface area Å 2 Q heat J q loadablty kg m -3 q molecular surface Å 2 R excess molar fracton R volume parameter r molecular dameter Å S entropy J K -1 T talng factor T energy functonal J mol -1 T temperature K t tme s U nternal energy J V external potental V volume m 3 V number v molar volume cm 3 mol -1 W work J w mass rato x molar rato y concentraton mol l -1 Greek symbols α selectvty absolute error screenng energy J mol -1 dfference δ relatve error γ actvty coeffcent ε delectrc constant C 2 J -1 m -1 ε porosty λ adjustable parameter wthn the COSMO-RS model µ chemcal potental J mol -1 µ moment s

14 v Lst of symbols ρ densty kg m -3 ρ electron densty e Å -3 σ charge densty e Å -2 σ surface tenson mn m -1 σ varance s τ parameter wthn the COSMO-RS model φ Θ ϕ ψ molar phase rato angle fugacty coeffcent wave functon ω wdth s Indces aac ads cav comb dsp don E eff exp ext hb HK nt j kn mn msft mob n OW acceptor adsorbent cavty combnatoral dspersve donator excess effectve expermental external hydrogen bondng Hohenberg - Kohn component nternal component knetc mnmal msft wthn the COSMO model moble phase runnng ndex octanol-water

15 v P partcle pot potental R elute res resdual sat saturaton sol sold stat statonary phase tot total w water I,II,III phase descrpton α, β phase descrpton nfnte dluton * deal 0 standard state Lst of symbols Constants e elementary charge e = 1,602177*10-19 C R unversal gas constant R = 8,3144 J mol-1 K-1 Abbrevatons COSMO Conductor-lke Screenng Model COSMO-RS Conductor-lke Screenng Model for Real Solvents CSM Contnuum Solvent Model DSC Dfferental Scannng Calormetry DFT Densty Functonal Theory FTIR Fourer transformed nfrared spectroscopy HF Hartree Fock LFER Lnear Free Energy Relatonshps LLE Lqud-Lqud Equlbrum LSER Lnear Solvaton Energy Relatonshps NMR Nuclear Magnetc Resonance QSAR Quanttatve Structure Actvty Relatonshps QSPR Quanttatve Structure Property Relatonshps QSRR Quanttatve Structure Retenton Relatonshps RMS Root-Mean-Square

16 v UNIFAC VLE UNIQUAC Functonal-group Actvty Coeffcents Vapour-Lqud Equlbrum Lst of symbols

17 1 Introducton 0 Enletung De Chromatographe wurde bezechnet als ene glechmäßge Perkolaton ener Flüssgket durch ene Säule bestehend aus mehr oder wenger fen geglederter Substanz de, mt welchen Mtteln auch mmer, bestmmte Flüssgketskomponenten retardert. [Martn 1950a] (snngemäß übersetzt). Dese frühe Defnton beschrebt en Verfahren, welches mttels enes Trennprozesses zwschen zwe Hlfsphasen n der Lage st, zwe oder mehrere Komponenten aus ener homogenen Mschung aufzutrennen. De ene Hlfsphase wrd als statonäre Phase bezechnet und besteht aus ortsgebundenen, festen oder flüssgen Komponenten, während de so genannte flüssge Phase dazu dent, de zu trennenden Komponenten mt Hlfe enes gasförmgen, flüssgen oder überkrtschen Fludstroms zu transporteren. De statonäre Phase st ncht unbedngt mt dem n der chromatographschen Säule gepackten Feststoff glechzusetzen. Oft tragen de n der Säule befndlchen Partkel de egentlche statonäre Phase auf hrer Oberfläche; z.b. n Form enes Flüssgketsflms oder kovalent gebundener Alkylketten. In der Umkehrphasen Hochdruck-Flüssgchromatographe (RP-HPLC) setzten sch de Hlfsphasen aus ener flüssgen (moblen) Phase und ener aus oberflächengebundenen Alkylketten bestehenden Festphase (statonäre Phase) zusammen. De Substanzen (Eluenten) welche es aufzutrennen glt, werden n der moblen Phase gelöst und so durch ene dchte Partkelpackung transportert. Ungeachtet der zugrunde legenden molekularen Mechansmen basert jeder Trenneffekt auf Thermodynamk, was sch n desem Fall als Untersched m Vertelungsverhalten zwschen den Eluenten manfestert. Aufgrund verscheden starker Wechselwrkungen verwelen Eluenten verscheden lang n der statonären Phase, was wederum zu unterschedlchen Elutonszeten und somt zur Auftrennung ener Mschung führt. Umkehrphasen-HPLC Systeme fnden zunehmende Bedeutung n velfältgen Anwendungen zur Analyse und Aufrengung verschedenster Stoffgruppen. Herzu zählen pharmazeutscher Wrkstoffe, Produkte der Lebensmttelndustre, ndustrelle Polymere und Fette, sowe Protene für Lfe-Scence Anwendungen. Schätzungen zufolge werden bs zu 70 % aller analytschen Trennungen nedermolekularer Proben mt Hlfe deser Technk durchgeführt [Neue 1997]. Aufgrund deser Anwendungsvelfalt und aus dem Bedürfns heraus ene Trennaufgabe mt überschaubarem Ensatz von Aufwand und Zet zu bewältgen wurde de Idee geboren, verschedene Herausforderungen mt computergestützter Systemsmulaton anzugehen.

18 2 Introducton Zu desen Herausforderungen gehören: () de Identfkaton enes auf das Eluentengemsch optmerten Trennsystems, () de Vorhersage enes trennsystemabhänggen Eluentenverhaltens und () de Identfzerung von Eluentenpeaks m Chromatogramm. Trotz der Velsetgket und der breten Palette an Ensatzmöglchketen basert der chromatographsche Trennprozess auf komplexen und n velen Telen noch unverstandenen Mechansmen. Aus desem Grund kann man de Modellerung enes solchen Mechansmus als durchaus komplexe Herausforderung bezechnen. Nach Guochon et al. [Guochon 2002] st de genaue Vorhersage des Adsorptonsglechgewchtes ene der ungelösten Aufgaben m Gebet der chromatographschen Forschung. Das COSMO-RS Model sowe de QSRR Methoden bezechnen zwe unterschedlche Herangehenswesen um chromatographsche Trennprozesse zu modelleren. De ene Möglchket beruht auf theoretschem dem Verständns physko-chemscher Prozesse und führt so zu fundamentalen thermodynamschen Bezehungen. De andere Herangehenswese basert auf der Anpassung systemspezfscher, emprscher Parameter und fndet berets wete Anwendung m Hnblck auf de Vorhersage chromatographschen Trennverhaltens. In der vorlegenden Arbet sollen bede Herangehenswesen untersucht werden, wobe herbe das Hauptaugenmerk auf der COSMO-RS baserten Methode legen wrd. Deser auf statstscher Thermodynamk fußende Vorhersageweg wrd grundsätzlch erst durch de Annahme ener pseudo-flüssgen statonären Phase gangbar. Dese Annahme kann als zentrale Hypothese der vorlegenden Arbet verstanden werden und soll den Weg fre machen n Rchtung ener a-pror Vorhersage von chromatographschem Retentonsverhalten.

19 3 Introducton Introducton Chromatography s the unform percolaton of a flud through a column of more or less fnely dvded substance, whch selectvely retards, by whatever means, certan components of the flud [Martn 1950a]. Ths defnton of chromatography pctures a technque that uses a separaton process between two auxlary phases to separate compounds from a homogenous mxture; one phase s called statonary beng sold or lqud and the other phase s denoted as moble because t s meant to transport the compounds to be separated n a gaseous, lqud or supercrtcal state. The statonary phase s not nevtably dentcal wth the sold packng of the chromatographc column. In an ncreasng number of applcatons, tghtly packed porous partcles (support) hold on ther surface the actual statonary phase e.g. n form of a lqud flm or a layer of covalently bound alkyl chans. In Reversed Phase Hgh Pressure Lqud Chromatography (RP-HPLC), the statonary phase s made up of alkyl chans covalently bound to sold partcles whle the moble phase conssts of a lqud. The substances that are meant to be separated (elutes) are dssolved wthn the moble phase whch then conveys them through a column of tghtly packed partcles. Regardless of the underlyng molecular mechansm, any separaton effect bases on thermodynamcs. In the present case, ths effect can be reduced to the dfference n elute parttonng between the two phases. Due to dfferent nteracton strengths, some elutes are retaned less strongly and therefore elute sooner that others. RP-HPLC fnds ts essental and versatle applcaton n the analyss and purfcaton of a very dverse set of substances, such as pharmaceutcals, products of the food ndustry, ndustral polymers, peptdes and protens for lfe-scence applcatons. It s beleved that up to 70 % of all analytcal separatons of low molecular samples are carred out usng ths method [Neue 1997]. Because of such a broad applcaton spectrum and the necessty to solve separaton tasks wth reasonable expermental effort and tme expense, system modellng s used as a method to tackle dvers objectves n the context of elute separaton as there are () the dentfcaton of deal separaton condtons relatve to a gven elute mxture, () the predcton of system specfc elute behavour and () the dentfcaton of resultng chromatographc peaks [Rethnger 2011]. Despte ts great varety and optons of applcaton, the chromatographc separaton process s complex and underlyng mechansms are not yet fully resolved. Therefore, modellng of such a process can be consdered as a rather dffcult task. Accurate predcton of the adsorpton equlbrum s one of the unsolved questons n the area of chromatographc research [Guochon 2002].

20 4 Introducton The COSMO-RS model and the Quanttatve Structure Retenton Relatonshps QSRR represent two dfferent groups of approaches that can be used to model chromatography: () one s based on the theoretcal understandng of physco-chemcal processes, leadng to the establshment of fundamental thermodynamc relatonshps. () The so far most popular way to predct retenton s based on large emprc coeffcent databases obtaned prmarly from expermental measurements that are then used to extrapolate towards unknown systems. Wthn the work at hand, both ways, () and () wll be nvestgated whle the man focus wll le on the former approach. The thermodynamc predcton path can be taken due to the assumpton of a pseudo-lqud statonary phase whch also represents the central hypothess of ths work. Ths pseudo-lqud approach shall open the door towards a-pror predcton of the chromatographc partton coeffcent and ncreased predcton qualty.

21 5 Goal of work 1 Goal of work Goal of the work at hand s, to apply the quantum chemstry and statstcal thermodynamcs based COSMO-RS model for the frst tme onto the predcton of chromatographc system separaton behavour. Predcton of lqud-lqud equlbra (LLE) [Maassen 1996]; [Clausen 2000] as well as of vapor-lqud equlbra (VLE) [Spuhl 2006] usng COSMO-RS has already become a standard of technology and shall now be expanded towards predcton of the chromatographc partton coeffcent K. The exstng COSMO-RS calculaton methodology must therefore be expanded n a way that enables equlbrum parttonng calculaton of an elute between a lqud moble and a complex statonary phase. Due to fact that most analytcal separatons of low molecular samples are carred out usng RP-HPLC systems, the man focus of expermental nvestgaton wthn ths work s lad on reversed statonary phases. To facltate applcaton of COSMO-RS onto complex RP-HPLC systems, a rather basc approach has been chosen. Wthn ths approach, the statonary phase s regarded as pseudolqud phase. Ths strategy was motvated from fndngs that chromatographc and lqudlqud partton coeffcents wll correlated over a wde range of values [Tsukahara 1993]. In a second step t wll be nvestgated f ths predcton approach can be appled onto socalled normal phase (NP-HPLC). Besdes the just mentoned approach, the COSMO model offers another possblty of predctng the behavour of a separaton system. Here, so-called molecule specfc σ-moments (structural descrptors) are generated on the mere bass of molecular structure. σ-moments can path a way towards descrpton of molecular nteractons by usng Quanttatve Structure Property Relatonshps (QSPR) [Klamt 2001]. Wthn ths work, t wll be nvestgated f QSPR usng σ-moments s an approach ft to pcture parttonng behavour wthn HPLC systems. Roundng off ths work and based on extensve lterature research, a schematc overvew shall be gven that ams to elaborate the varety and nterrelatons of predcton methods for the separaton behavour of chromatographc systems.

22 6 Bascs 2 Bascs The sectons below are meant to gve an overvew on the current state of knowledge and technology. The followng sectons shall equp the reader wth the bascs of chromatography and ts modelng. Frst, the bascs of chromatographc separaton wll be explaned and the reader wll be acquanted to correspondng termnology. A second part wll gve an overvew on chromatographc phases and methods of phase nvestgaton, whle part three explans bascs of phase equlbrum. Part four wll lst and shortly explan approaches that have been used to model chromatography. The last part of Secton 4 s meant to gve nsghts nto chemcal and quantum chemcal bascs on whch most calculatons wthn ths work are based on. 2.1 Bascs of chromatography In the followng, basc parameters, equatons and correlatons, requred to characterze a chromatographc separaton systems are presented Chromatographc separaton prncple Chromatography s a unt operaton that uses a separaton process between two auxlary phases to separate two or more compounds from a homogenous mxture; one s called statonary phase beng sold or lqud and the other moble flud phase transportng the compounds that may consst of gaseous, lqud or supercrtcal state. The statonary phase s not necessarly dentcal wth the sold packng of the chromatographc column. In many cases, tghtly packed porous partcles can act as support and hold on ther surface the actual statonary phase, e.g. n form of a lqud flm or a layer of covalently bound alkyl chans. To undergo nteractons wth the percolatng compounds, the statonary phase has to exhbt the proper functonal groups. Dependng on the ndvdual nteracton strength, a compound wll preferably resde n moble or statonary phase. Compound eluton speed s therefore drectly dependng on ts resdence probablty between the two phases. From here, compounds that percolate through a chromatographc system wll be termed as elutes. Fg. 2.1 pctures elute separaton behavour nsde a chromatographc column by llustratng snapshots n tme.

23 7 Bascs Fgure 2.1: Separaton prncple n chromatography Frst, a sample consstng of three dfferent elutes s ejected nto the moble phase stream. Then, dependng on dfferences n ther nteracton strength wth the statonary phase, ndvdual partton equlbra wll lead to a contnuous gap ncrease between the elutes. At the end of the column, due to ther dsplacement n space, the elutes wll leave the system consecutvely Volume and porosty n a chromatographc column The total volume nsde a chromatographc column can be broken up nto four dfferent parts: () the volume between the porous statonary phase partcles V ext, () the pore volume of the statonary phase partcles sum of partcle and pore volumes V P. V nt, () the sold partcle volume wthout pores V sol and (v) the Fgure 2.2: Fractonal volumes nsde a chromatographc column From these volumes, dfferent porostes can then be defned [Sedel-Morgenstern 1995]. The rato of moble phase volume and total column volume s called total porostyε tot and s gven by the followng equaton:

24 8 Bascs V = + V ext nt ε tot (2.1) VC Whereas external or ntersttal bed porosty ε ext s defned as the rato of the nterpartcle volume V ext and the column volumev C. V = ext ε ext (2.2) VC The rato of ntrapartcle pore volume nternal porostyε nt. V nt and partcle volume V P s n turn defned as the V nt nt = V P ε (2.3) The total and external porosty are lnked va the nternal porosty by the followng equaton: ( 1 ε ) ε nt ε tot = ε ext + ext (2.4) Total as well as external bed porosty can be determned from experments usng dfferent tracer molecules. To assess total porosty, a tracer must be small enough to enter the pore of the statonary phase partcles wthout nteractng wth surface groups. A non-nteractng molecule structure, large enough and therefore stercally hndered to enter the pores can be used as tracer to determne the external porosty. If V nt s consdered to be part of the moble phase volume, the fractonal moble phase volume ε equals the total porostyε tot depct the external porosty.. If consdered to be part of the sorbent phase, ε wll The volume rato of statonary and moble phase s commonly referred to as phase rato F, whle the molar phase rato wll n ths context be referred to as Φ. F can be expressed n terms of the fractonal volumeε. F = 1 ε (2.5) ε Retenton tme and related quanttes Eluton tme or retenton tme t R, s defned as the tme span between the tme of njecton and the pont n tme, when half the mass of the njected elute has eluted from the column. In the

25 9 Bascs followng, solute molecules that elute from a chromatographc system wll be referred to as elute. To record eluton tmes and separaton qualty, the moble phase wll pass a detector after havng eluted through the column. The concentraton of moble phase dssolved components wll be detected over tme, leadng to a sgnal-over-tme recordng, the so-called chromatogram. The deflectons correspondng to the detected components are referred to as peaks. Fgure 2.3: Chromatogram for the pulse njecton of a four component mxture contanng three retaned and one unretaned elute t R1, t R2 and t R3 depct the correspondng retenton tmes of elutes 1,2 and 3, whle t 0 stands for the column dead tme. Bascally, column dead tme t 0 s defned n accordance to t R,. But an elute, ft to measure t 0 s requred to not have any nteractons wth the statonary phase. Dependng on ndvdual sze, the accessble column volume can vary for dfferent elutes. To determne the external dead tme t 0,ext a tracer substance can be used whch s stercally hndered to penetrate the statonary phase pore volume, whle pore penetratng molecules lke the pyrmdne dervatve uracl can be used to determne the nternal dead tme t 0,nt [Schulte 2005].

26 10 Bascs In case of a symmetrcal peak on the chromatogram, t R, s the tme span between elute njecton and peak maxmum of the correspondng detector sgnal. For asymmetrcal peaks, the apex of the peak wll not concde wth the pont n tme, where half of the component mass has eluted thought the column. Therefore retenton tme t R, wll be determned by the frst moment of the peak µ 1,. 0 ( t) t dt µ = (2.6) 1, c 0 c ( t) dt c stands for the detected concentraton of elute. The use of retenton tme to descrbe a certan chromatographc system suffers from the dsadvantage of dependng on moble phase flow velocty [Schulte 2005]. To overcome ths dependence, retenton data s mostly gven n terms of a dmensonless rato between net retenton tme (t R, t 0 ) and column dead tme t 0, the so-called capacty factor k, retenton factor or k-factor. k ' t R, 0 = (2.7) t t 0 Dependng only on the elute dstrbuton between the two auxlary phases, k s defned as a purely thermodynamc parameter. As a dmensonless rato of the net retenton tmes of two elutes and j, the selectvty or separaton factor α j s ntroduced. α j can be expressed as a rato of the partton coeffcents K (Eq. 2.8). The separaton factor gves nformaton on whether a separaton s possble (for α j 1) from a purely thermodynamc pont of vew. K t t αβ, 0 α R j = = αβ (2.8) K j t R, j t0 A hgh separaton factor means that the chromatographc peaks can be dstngushed, but they mght stll overlap due to ther broadness. Therefore a satsfactory separaton result s not guaranteed wth ths thermodynamc parameter.

27 11 Bascs Peak wdth and related quanttes Peak wdth ω expresses the peak broadenng wtch wll take place durng the elute eluton through a column. Fgure 2.4: Mechansm leadng to chromatographc peak broadenng Allowng for conclusons on separaton system effcency, peak wdth s another mportant parameter n peak descrpton. Dfferent postons relatve to the peak heght can be used to determne the peak wdth. Most commonly, peak wdth s beng measured at 10% and at 50% peak heght, whch wll then be gven n terms of ω,0.1 and ω,0.5 respectvely. Another method to descrbe peak spreadng s to use the second central moment, whch s dentcal to the varance σ ² of the peak. In analogy to the frst moment, σ ² s calculated ndependent from a prorly chosen poston. 2 c ( t µ ) dt 2 0 σ = (2.9) 0 1, c dt The talng factor T s meant to descrbe the degree of asymmetry of a peak. It s calculated by the rato of the correspondng wdths of the two peak halves a and b at 10 % peak heght, wth the peak beng dvded at ts apex poston. b,0.1 T = (2.10) a,0.1 Regardng the effectveness of the entre chromatographc separaton system, the resoluton R S can be consdered as a well ftted measure. Beng calculated from dfference n retenton tme and peak wdths, t combnes thermodynamc as well as effcency related elements.

28 12 Bascs R S = 2 ( t t ) R, j ω ω R, j (2.11) Wth ω and ω j beng the component base lne peak wdths. Another parameter that s used to evaluate chromatographc systems s the plate number N. In 1941, Martn and Synge [Martn 1941] modelled a chromatographc column as a cascade of N deally strred plates or tanks. These days N s also referred to as column effcency. For dfferent elutes, N vares for a constant system. Wth symmetrcal peaks, the effcency N can be calculated by the followng equaton: N ( ω ) 2 = (2.12) t R, In general, N can be calculated as a rato of the frst absolute and second central peak moment. N µ = (2.13) σ 2 1, The moble phase The choce of the moble phase can be vewed as a frst step n the development of a separaton system. The moble phase conveys the elutes past or through the porous statonary bed of a chromatographc column. Most commonly a mxture of dfferent solvents wll be used to obtan an optmum n separaton result. If a moble phase s to be chosen, accordng to [Schulte 2005], partcularly four system qualtes have [Lottes 2009] to be taken nto account: () throughput, () stablty, () safety concerns and (v) operatng condtons.

29 13 Bascs The statonary Phase Statonary phase nteractons wth elute and moble phase play a defnng role n elute retenton. Snce the development of frst LC applcatons, dfferent materals have been found to be applcable as statonary phases. These dfferences n statonary phase materal have led to a varety n separaton methods (Tab. 2.1). Table 2.1: Separaton method classfcaton based on dfferences n statonary phase materal Separaton Method Phase materal Annotaton Normal phase chromatography (NP-HPLC) Partton chromatography Reversed phase chromatography (RP-HPLC) Sze excluson chromatography (SEC) Ion exchange chromatography Affnty chromatography Slca gels, Alumnum oxdes Lqud flm on a sold carrer materal Chemcally modfed slca gels Cross lnked polystyrene, slca Ion exchange resn, carryng charged functonal groups Gel matrx (e.g. agarose) Usually assocated wth adsorpton chromatography (see Secton 2.2.7) Retenton s based on dfferences n elute solublty Functonal surface groups bound to the partcle surface (see Secton 2.1.7). Due to bonded phase nhomogenety, bound actve groups and other effects, the retenton mechansm s not yet fully understood. Usng homogenous partcle sze- and pore-wdth dstrbuton to facltate non nteractve elute sze separaton. A charged statonary phase wll retan oppostely charged elutes A hghly specfc bologcal nteracton (.e. antgen / antbody nteracton) wll retan target elute molecules Due to ts benefcal characterstcs, slca gel s the most commonly used materal n chromatography. Ths fact can be traced back to ts applcaton as carrer materal, where t serves as bass for many chemcally modfed statonary phases. In pure state t s used n NP- HPLC or sze excluson chromatographc applcatons (Tab. 2.1).

30 14 Bascs Fgure 2.5: Slca spheres before (a) and after (b) sze classfcaton [Unger 1990] Slcagel (Fg. 2.5) conssts of slca atoms beng three-dmensonally lnked va oxygen atoms. On ts surface, the gel s saturated wth so-called slanol groups. In normal phase HPLC applcatons, these groups serve as adsorptve centres. For altered slca gel versons, slanol groups act as lnk for chemcal modfcaton. Due to an amorphous character and ts heterogeneous surface, t s a challenge for the numerous manufacturers on the market to produce well-defned slca gel partcles. In chromatography, the statonary phase s always packed nto a column. Ths desgn s consdered as the core tem of chromatography and can be characterzed by a number of parameters (Tab. 2.2). Table 2.2: Statonary phase parameters Statonary phase parameter Specfc surface (m²/g) Partcle shape Partcle sze (µm) Partcle materal Pore wdth Pore wdth dstrbuton Column length and nner dameter Annotaton Wth decreasng surface, the k -factor wll also decrease Sphercal or non-sphercal. In general sphercal partcles wll show better separaton performance [Lottes 2009] Most common partcle dameters n analytcal chromatography: 3µm, 5 µm, 7 µm, 10 µm Column effcency approxmately doubles wth each step towards smaller dameter Most commonly sold gels (e.g. slcagel). Others: glass beads, cross-lnked polystyrols, on exchange resns or porous graphte Exact pore wdth s mportant for sterc separaton Narrow pore wdth dstrbuton wll lead to more symmetrc peaks Column effcency wll change dsproportonate to column length

31 15 Bascs The choce of chromatographc columns on the market s almost unmanageable. Every company wll use ther own brand names and statonary phase or column parameters. An ad n fndng a proper column s gven by the Unted States Pharmacopea (USP). USP lstngs are sorted by statonary phase materal and not by company name. It provdes a cumulatve lstng of columns referenced n gas- and lqud-chromatographc methods The reversed statonary phase Orgnally, covalently bonded reversed phase packng materals were ntroduced to combne two characterstcs: () stablty of a lqud-sold chromatographc system whle () exhbtng the absorpton (parttonng) behavour of a lqud-lqud system [Snyder 1979]. Snce that tme, a dscusson of how to represent elute dstrbuton between moble and statonary phases has been gong on (Secton 2.2.6) [Snyder 1968]; [Melander 1980]; [Jaronec 1982]; [Jaronec 1985]; [Sander 1987]; [Dorsey 1989]; [Unger 1990]. The great majorty (more than 70%) of all applcatons used n the vast feld of lqud chromatography (LC) employ statonary phase partcles covered wth covalently bonded lgands. Prmarly due to ts ablty for chemcal modfcaton, slca s partcularly useful as base materal for the desgn of modfed separaton meda n chromatography. To modfy slca towards a reversed phase partcle, predomnantly organoslanzaton s appled. Hereby, a surface reacton wll covalently bnd organoslane molecules to the slca surface [Unger 1990]. Most stable are slca gels wth functonal groups bound to the surface va S-O-S-C-R bonds, whle usng mono- or dchlorslanes for chemcal converson. Most reversed phases used wthn ths work (e.g. C18) are produced by applcaton of ths type of chemcal reacton. The surface reacton usng organoslanes can be wrtten as follows: -SOH + X-SR3 -S-O-SR3 + HX The bonded phase resultng from chemcal modfcaton wll provde the specfc surface character. Surface bndng can be accomplshed by a monomerc or a polymerc approach. The latter wll not only create one attachment pont between slca surface and organoslane reagent but also cross-lnk the bonded phase by sloxane lnkage [Cazes 2010]. Due to ther cross-lnked network, polymerc statonary phases are more stable and resstant to hydrolytc degradaton when n contact wth aqueous moble phases, whle monomerc statonary phases offer the hgher separaton effcency.

32 16 Bascs A selecton of alkyl groups, whch have been used as reversed phase materals wthn ths work are shown n the followng table. Table2.3: Alkyl groups used as reversed phase lgands Abbrevaton Name Structure C1 Methyl -S-O-S-CH 3 C8 Octyl -S-O-S-(CH 2 ) 7 -CH 3 C18 (ODS) Octadecyl -S-O-S-(CH 2 ) 17 -CH 3 Phenyl Phenyl -S-O-S -(CH 2 ) 3 -C 6 H 5 CN Cyano -S-O-S-(CH 2 ) 3 -CN In reversed phase separaton systems, structural effects of the bonded phase have great mpact on retenton. As lgands can dffer n ther bondng densty, length, conformatons, orentaton, dynamcs and actve groups, the nature of alkyl bonded phases s rather complex. Nether lqud phase nor sold phase models are capable to exactly represent these nhomogeneous phases [Unger 1990]. The followng secton s therefore meant to gve an overvew on methods and technques that have been used to resolve the complexty and entangle the superposton of the many nfluencng parameters The reversed statonary phase: Technques of examnaton Although RP-HPLC separaton technques are popular and wdely used n analytcal just as n bgger scale preparatve applcatons, flow structure, lgand behavour and nature of molecular nteractons on a mcroscopc level are stll not resolved thoroughly. Because of ts complexty, the lnk between macroscopc effects and ts mcroscopc account has not yet been suffcently establshed. To approach the physcs of separaton from a theoretcal bass, relable nformaton about mcro scale dynamcs and structural condtons needs to be accessble. In the followng, a bref revew on a number of nonnvasve expermental technques, ncludng NMR spectroscopy, FTIR spectroscopy, Dfferental Scannng Calormetry (DSC) and Raman spectroscopy wll be gven. For more detaled nformaton see the revew paper of Sander et al. [Sander 2005]. These technques are capable to provde more drect evdence of bonded phase character n terms of lgand moton, conformaton, and cooperatve assocatons.

33 17 Bascs NMR In 1938 Nuclear Magnetc Resonance (NMR) was frst observed and 14 years later the Nobel Prze n physcs was gven to F. Bloch and E. M. Purcell for ther poneerng work n NMR technque. In the presence of a statc magnetc feld and a second oscllatng magnetc feld, some atomc nucle wll exhbt specfc quantum mechancal magnetc propertes. Ths phenomenon s called NMR. Although all nucle that posses a spn are subject to NMR, analyss of nucle havng a fractonal spn state (e.g. spn = 1/2) s qute straghtforward. Therefore the most preferred sotopes for NMR spectroscopc measurements are 1 H, 13 C, 19 F and 31 P. NMR spectroscopy represents a powerful tool to obtan detaled physcal, chemcal, electronc and structural nformaton about molecules n soluton and n sold state. Due to ths non-nvasve analytcal potental, NMR spectroscopy has become an essental tool for characterzaton of statonary phase materals under varyng chromatographc system condtons. Partcularly sold state NMR-Technques, lke 29 S NMR, 13 C-NMR, 1 H-NMR [Fatunmb 1993]; [Scholten 1996], 19 F NMR [Kamlet 1976b]; [Kamlet 1983], 13 C CP/MAS NMR [Pnes 1973] or 2 H T 1 NMR [Wysock 1998] have proven capable to nvestgate structural propertes and behavour of bound alkyl lgands n RP-HPLC phases. The relatonshps between chromatographc propertes and statonary phase surface propertes lke the nfluence of shelded and accessble resdual surface slanols [Scholten 1997]; [Scholten 1994]; [Vansant 1995] or the structural confguraton of the bonded alkyl phase [Buszewsk 1997]; [Shah 1987]; [Haeberlen 1996; Scholten 1996]; [Kobayash 2005]; [Macel 1980]; [Sndorf 1983]; [Sander 1995]; [Buszewsk 2003]; [Buszewsk 2006]; [Bruch 2003]; [Pursch 1996a] were nvestgated. So was lgand bondng densty [Berezntsk 1998]; [Buszewsk 2006]; [Glpn 1984]; [Shah 1987]; [Bayer 1986]; [Srnvasan 2006b]. All structural parameters mentoned above are nvolved n determnng the conformatonal order of the alkyl chan moetes and are therefore ntmately lnked wth the selectvty durng chromatographc separatons. Apart from structural confguraton, external parameters nfluence the conformatonal order. System temperature [Jnno 1989]; [Kelusky 1986]; [Gangoda 1983]; [Thompson 1994]; [Srnvasan 2006b] just as the nfluence of dfferent moble phases [Blesner 1993]; [Zegler 1991b]; [Zegler 1991a]; [Kelusky 1986]; [Ducey, Jr. 2002a]; [Orendorff 2003]; [Marshall 1984] have been subject to several NMR nvestgatons. Other authors used NMR methods on questons of elute mgraton and transport nsde statonary phases [Tallarek 1998]; [Veth 2004]; [Zegler 1991b] or amed to valdate the assumpton of lqud-lke behavour of bonded phase alkyl chans [Albert 1990]; [Albert 1991]. NMR spectroscopy s found to be a very versatle tool to study the statonary phase n

34 18 Bascs terms of structure and dynamc behavour and mght lead to a better understandng of the chromatographc process FTIR In the 1960s, Snyder showed the applcablty of nfrared spectroscopy (IR) to alkyl chan conformaton nvestgaton n pure alkanes and model membranes [Snyder 1967]. Some years later, FTIR (Fourer transform nfrared spectroscopy) was developed from conventonal IR provdng a number of advantages as ncreased senstvty, speed and mproved data processng. Sander et al. were frst to use ths technque on a study of alkyl chan conformatons. Usng FTIR, qualtatve data concernng changes n statonary phase conformatonal order can be ganed from the symmetrc and ant-symetrc stretchng band maxma postons of tethered CH 2 groups. In the followng, the nfluence of dfferent parameters on alkyl chan conformatons has been nvestgated by several authors: () Temperatur [Srnvasan 2004]; [Srnvasan 2005a]; [Srnvasan 2006b]. Wth reduced temperature and ncreasng alkyl chan densty or length, Sgh et al. [Sngh 2002] found an ncreased conformatonal order of bonded chans. () Pressure [Srnvasan 2005b]. () Surface coverage [Sngh 2002]; [Srnvasan 2006b]. (v) Alkyl chan length and poston [Sngh 2002]. (v) Solvent nfluence [Srnvasan 2006a] Raman spectroscopy In 1928, Chandrasekhara Venkata Raman found that rrtated molecules addtonally scatter lght other than the orgnatng monochromatc lght source frequency. Devatons n emtted lght spectra can be assgned so specfc molecule structures. Therefore each materal wll gve a unque spectral fngerprnt. Based on ths effect, Raman spectroscopy has been developed. After the potental of Raman spectroscopy to detect dfferent states of order n lpd was found by Larsson n 1973 [Larsson 1973], the ntensty rato of the ant-symmetrc and symmetrc methylene bands as well as the frequency of assocated Raman bands, were related to conformatonal order. In vew of bonded lgand conformatons n RP-HPLC, Thomson et al. showed the applcablty to slca bonded alkylslane layers [Thompson 1994] and soon Raman spectroscopy was appled to nvestgate conformatonal order of bonded alkyl lgands under varous condtons. Ho et al. [Ho 1998] was frst to nvestgate the temperature effect on polymerc and monomerc C18 phases. Along wth other authors, they observed sgnfcant temperature nfluence on statonary phase conformatonal order [Pursch 1996b]; [Ducey, Jr. 2002b].

35 19 Bascs In contrast to FTIR methods, Raman spectroscopy does not suffer from scatterng, absorbed water or slca nterferences Neutron scatterng Neutron scatterng s another spectral characterzaton approach to obtan data on bonded phase thckness [Pursch 1999], volume fracton, chan conformaton and moton [Beaufls 1985]. To obtan nanometer scale refractve ndex nformaton, a neutron beam s drected at a sample. By nteractng wth sample atom nucle, the neutrons are elastcally scattered leadng to an elastc scatterng peak broadenng. E.g. Sander et al. [Sander 1990] used small angle neutron scatterng experments to obtan bonded phase thckness of about 17 A for monomerc C DSC Dfferental Scannng Calormetry s an nstrument for thermal analyss. It s used to estmate the amount of heat generated by or appled to a physcal or chemcal substance converson. Therefore DSC s capable to resolve phase transtons that occur n system phases.e. RP- HPLC bonded phases [Claudy 1985]; [Morel 1987]. An early attempt usng DSC to examne behavour of bonded C18 and C22 alkyl lgands was done by Hansen et al. [Hansen 1983]. Whle for pure C18 an endothermc phase transton was found at 31 C, no dstnct phase transtons were found for bonded C18 lgands of ntermedate bondng denstes ( µmol/m 2 ). Other authors found weak transtons for polymerc C18 phases at temperatures above 35 C [Jnno 1988]. Although gvng thermodynamc nformaton about the bonded phase, DSC does not seem capable to delver drect conformatonal state nformaton Macroscopc vew on statonary phase As expermental technques am to resolve retenton mechansms on a molecular scale, wth a vew to advance HPLC smulaton models, the neglect of non-unform packng structures lead to dscrepances between the model forecast and the expermental fndngs whle scalng up chromatographc columns [Heuer 1996]. The fact that from a macroscopc pont of vew, chromatographc columns possess heterogenetes s known to chromatographers snce a long tme [Baur 1988]; [Tallarek 1995]. Expermental evdence that the packng structure n chromatographc columns s not necessarly homogeneous was summarzed n a revew artcle by Guochon et al. [Guochon 1997].

36 20 Bascs Modern magng technques lke MRI (Magnetc Resonance Imagng) [Labln 2007] or x-ray CT (X-ray computed tomography) [Astrath 2007b] were found to be sutable for characterzng these heterogenetes n more detal, e.g. axal and radal porosty dstrbutons. Expermental work by Lottes et al. [Lottes 2009] showed that the shape of the statonary phase materal can have strong nfluence on the flow profles and therefore on the overall performance of the LC process. The reasons for heterogeneous regons n the packed bed are further to be studed but wll be most lkely a result of the frcton between the packng materal and the column wall durng the packng process [Yew 2003]. 2.2 Bascs of phase equlbra The frst law of thermodynamcs expresses that energy can be transformed from one form nto another but t cannot be created or destroyed. Consderng rgd body mechancs, the law of energy conservaton can be wrtten as follows: E = E + E const. (2.14) kn pot = In Eq the external state of a system s represented by E kn and E pot depctng the knetc energy and the potental energy, respectvely. For thermodynamc treatment, also the nternal state of a system needs to be consdered. The followng equaton shows the 1 st law for a closed system and reversble processes states. du = δ Q + δw (2.15) The letter δ denotes a dfferental operator for a non-state property. A closed system can exchange energy wth ts surroundngs n form of heat Q and work W, leadng to a change n nternal energy U. Due to the second law of thermodynamcs, the entropy S of a system wll not decrease, except the entropy of some other system s beng ncreased. For the dealzed concept of an solated system, wthout enery transfer across ts boundares, ths would mply mpossblty of an entropy decrease. ds 0 (2.16) Entropy can be understood as a measure of the degree of a system organzaton or dsorganzaton or as a measure of the amount of energy (heat) n a physcal system that cannot be transformed nto thermodynamc work. The latter formulaton leads to the followng expresson for reversble processes:

37 21 Bascs dq ds = (2.17) T Ths extensve state varable was ntroduced by Rudolf Clausus n 1865 and confrmed wth statstcal mechancal means n 1880 by Ludwg Boltzmann. Josah Wllard Gbbs showed that a mnmum system nternal energy s equvalent wth ts sentropc equlbrum state accordng to Eq Ths correlaton s referred to as extremum prncple: ( U = ) S, V, n ( ) = const S = max U, V, n = const mn (2.18) To completely represent the thermodynamc state of a system, besdes the nternal energy U, other thermodynamc potentals or fundamental functons were defned. They all can be derved usng Legendre transforms from an expresson for U. Enthalpy H and Gbbs energy G (or free enthalpy) are two of these thermodynamc potentals. H U + PV (2.19) G H TS (2.20) The varables T and P depct system temperature and pressure, respectvely. Accordng to the 1 st law of thermodynamcs, t s possble to descrbe an open system whch allows exchange of matter as well as energy across ts boundares by the ndependent varables entropy S, volume V and molar amounts n 1, n 2, n r, where r s the number of components. The nternal energy U s consdered to be a functon of these varables. ( S V, n, n ) U = U... (2.21), 1 2 n r Buldng the total dfferental gves du U = S V, n U ds + V S, n U dv + n S, V n, j dn (2.22) wth n j referrng to all mole numbers other than the th. For the frst two dervatves of Eq. 2.22, followng denttes for a homogeneous closed system can be appled:

38 22 Bascs U S V = T (2.23) and U V S = P (2.24) Wth the defnton of the chemcal potental µ, U = µ n (2.25) S, V, n j Eq can be wrtten as follows: + du = TdS PdV µ dn (2.26) The chemcal potental µ depcts the amount of energy, whch enters or leaves the system va component. Eq s consdered to be the fundamental equaton for an open system or the so-called fundamental equatons of Gbbs. Applcaton of Eq on Eq and Eq leads to the fundamental equatons of enthalpy and Gbbs energy, respectvely: + dh = TdS + VdP µ dn (2.27) + dg = SdT + VdP µ dn (2.28) Lookng at the three fundamental equatons above, Gbbs energy offers a practcable way to descrbe a thermodynamc system by usng the varables temperature, pressure and system composton. Referrng to Eq. 2.29, the chemcal potental can also be expressed as a partal dervatve of the Gbbs energy G. G = µ n (2.29) T, P, n j The chemcal potental n an deal gas mxture can be quantfy as:

39 23 Bascs d d + P µ ( T, P) = µ 0 ( T, P ) + RT ln + RT ln y + (2.30) P The chemcal potental of an deal gas µ d s ftted to a reference state at the system temperature and an arbtrary reference pressure P +. The frst correcton term wthn Eq s used to adjust P + to system condtons, whle the second one accounts for the partal pressure P d by use of the molar compound concentraton y. In 1908, the concept of fugacty f was ntroduced by Glbert N. Lews [Lews 1908]. Ths quantty was meant to better descrbe real systems by substtutng the pressure P. Bascally fugacty depcts flud phase pressure but wth an addtonal consderaton of ntermolecular forces. Hence, for an deal system wthout molecular nteractons, values for fugacty and pressure become equal. The mentoned nteractons can be expressed by the fugacty coeffcent ϕ. Consderng the followng expresson, f y ϕ P (2.31) for a real system and a pure compound can then be wrtten: d + P f0 µ 0 ( T, P) = µ 0 ( T, P ) + RT ln + RT ln + (2.32) P P For descrpton of real lqud mxtures, the actvty a was ntroduced. a s defned as the rato of fugacty f of a mxture component and ts standard fugacty f 0. a f f = (2.33) 0 The dmensonless actvty coeffcent γ depcts the rato of a and any measure of concentraton. In lqud phase generally the mole fracton x s beng used. a = γ (2.34) x Therefore, compared to Eq. 2.32, another opton of chemcal potental calculaton wthn an non-deal system unfolds.

40 24 Bascs P µ = ln + P d + ( T, P) µ 0 ( T, P ) + RT ln + RT ln( x ) + RT ( γ ) (2.35) The last term of the above equaton depcts the so-called excess part, holdng an entropc and an enthalpc contrbuton. For an deal system, the actvty coeffcent wll have a value of one, causng the last term to drop out Thermodynamc equlbrum In a closed system, a pure substance or a mxture of several components are sad to be n thermodynamc equlbrum, f the nternal energy U of the system has reached a mnmum n the frame of ts fundamental varables. In that equlbrum state, the system can form a homogenous phase or exst n several coexstng phases. In case of coexstng phases, equlbrum state also leads to equlbrum concentratons of all components wthn all phases and no macroscopc matter exchange s measurable. Based on fxed temperature and pressure, equlbrum thermodynamcs gves nformaton on phase composton and number of coexstng phases. Equlbrum state s defned by equalty of temperature T, pressure P and chemcal potental µ of all components wthn all phases [Prausntz 1999]. In other words, ths can be expressed as a thermal, mechancal and chemcal equlbrum. In a system consstng of k components and π phases, the equlbrum relatonshp s expressed as follows, T P I I II ϕ π = T =... = T = T thermal equlbrum (2.36) II ϕ π = P =... = P = P mechancal equlbrum (2.37) I II ϕ π µ µ = = µ = µ =... chemcal equlbrum = 1... k (2.38) whle the rule of mxture of Gbbs wll quantfy k and π. F = 2 π + k (2.39) F depcts the degrees of freedom of the consdered system. If referrng all phases to the same standard state, Eq and Eq wll lead to the sofugacty crteron. I II ϕ π f = f =... = f = f sofugacty crteron (2.40) By ntroducng fugacty or actvty nto the sofugacty crteron, phase equlbrum relatonshps can be establshed. Flud and lqud phase equlbrum compostons can be

41 25 Bascs calculated usng these relatonshps. For a system consstng of several phases, followng equatons can be derved: I I II II y ϕ = y ϕ =... = y ϕ,ϕ-concept (2.41) π ϕ π I I II II x γ = x γ =... = x γ,γ-concept (2.42) π γ π I I II II x γ f 0 = x ϕ =... γ,ϕ-concept (2.43) Dependng on the type of phase equlbrum (vapour lqud equlbrum VLE or lqud lqud equlbrum LLE) and methods used to pcture ntermolecular forces, t s essental to choose the proper one of the concepts above. If appled on mxture phase equlbra wth accessble excess enthalpes and entropes, t s suggestve to utlze an actvty coeffcent based descrpton (e.g. the γ,γ-concept). Calculaton of partal free excess enthalpes g E s realzable by so-called g E -models (Secton 2.3.2). By combnng Eq and Eq t becomes obvous, that g E s representable by the actvty coeffcent γ. G n E T, P, n j = RT lnγ = g E = h E Ts E (2.44) Elute dstrbuton equlbrum n a chromatographc system Regardless of the underlyng molecular mechansm, every chromatographc process conssts of elute dstrbuton between moble and statonary phase and s therefore governed by thermodynamcs. After havng entered the column at a tme t = 0, deally at a tme span of 0 (Drac short pulse njecton), and mgratng through the column wth a constant flow rate, elute molecules contnuously commute between the moble and statonary phase. Dfferences n the tme span of whch a specfc elute wll resde n the statonary phase wll consequently lead to varyng eluton tmes between dfferent elutes. Elute separaton n chromatography s therefore facltated by dfferences n phase dstrbuton. In the followng, the term parttonng wll refer to compound dstrbuton between two volumes, whle dstrbuton apples to compound allocaton between two enttes n general. Therefore, an adsorpton mechansm wll lead to compound dstrbuton but not to compound parttonng.

42 26 Bascs Dstrbuton of compound between two phases can always be expressed n terms of ts partton coeffcent or equlbrum constant K. Assumng a system consstng of two phases, compound wll dstrbute accordng to the thermodynamc equlbrum. Therefore, the partton coeffcent characterzes the dstrbuton behavour of compound and t s defned as the rato of ts mole fractons x n phase α and phase β, respectvely. K αβ x = (2.45) x α β In some lterature, the partton coeffcent s defned as a rato of concentratons c. P αβ c = (2.46) c α β Consderng the defntons of mole fracton x, molar concentraton c and molar volume ν, x α n = α α n j (2.47) c α = α v = n V V n α α α α (2.48) (2.49) both defntons of the partton coeffcent can be lnked as follows: α α x n α α β β αβ c n V x ν αβ ν P = = = V = = K β β α α (2.50) c n V x n x ν ν α β α β β V β α β ν α und ν β depct the molar volumes of both phases. From standard thermodynamcs, the partton coeffcent at constant temperature T s related to the molar Gbbs energy change as: g = RT ln (2.51) 0 αβ K The temperature dependence of the partton coeffcent s gven by the general Gbbs- Helmholtz equaton expressed by K by the help of Eq. 2.51:

43 27 Bascs d ln K dt αβ = h RT 0 2 (2.52) Accordng to Eq. 2.7, k relates the tme span an elute s held by the statonary phase to ts resdence tme n the moble phase. In other words, whle a gven amount of a specfc elute molecule passes through a column, t wll spent a certan tme n moble phase and the remanng tme n statonary phase. Consderng ths rato of tme ntervals and ntegratng the amount of elute wth respect to ts habtaton over the system eluton tme, the capacty factor can also be expressed as a mole rato of elute n the two auxlary phases [Schulte 2005]. n k ' = (2.53) n stat mob Whle here α stands for the statonary phase and β for the moble phase, respectvely. Hence, k lnks the physcal molecular propertes of a compound to ts column retenton tme, makng t a major parameter n chromatography. As explaned n Secton 2.1.3, t 0 can vary wth the tracer molecule beng used. Consequently t 0 related values lke net retenton tme (t R, t 0 ) and should only be compared f usng the same tracer molecule. k -factor and partton coeffcent K αβ are lnked va the column phase rato φ. From Eq and the defnton for υ follows: K αβ β β n n ' n = = k α α (2.54) n n n α β Wth the so-called phase rato φ beng the rato of the total number of moles of statonary phase to the total number of moles of moble phase, α n φ = (2.55) β n t can be wrtten: αβ k ' = φ (2.56) K

44 28 Bascs The adsorpton equlbrum Fgure 2.6: Nomenclature of the adsorpton process Adsorpton s a surface effect and can be defned as an ncrease n the concentraton of a dssolved compound at the nterface of a condensed and a lqud phase due to the operaton of surface forces [IUPAC Gold Book 1997]. In general, ths physcal phenomenon can be seen as a compound concentraton dfference from bulk lqud to ts phase boundares [Hrsch 2000]. Such an accumulaton process wll create a molecule flm (adsorbate) on a surface materal (adsorbent) as can be seen from Fg As adsorpton s regarded as a surface effect, statonary phases n chromatography are requred to exhbt a surface, to allow for adsorpton as separaton prncple. Adsorpton chromatography would be defned as chromatography n whch separaton s based manly on dfferences between the adsorpton affntes of the sample components for the surface of an actve sold [IUPAC Gold Book 1997]. Lookng at statonary phases where sold partcles exhbt a dstnct surface lke slca partcles do n NP-HPLC (Normal Phase Hgh-Performance Lqud Chromatography), separaton prncple classfcaton s a smple task. On the other hand, when RP-HPLC wth ts slca bonded alkyl chans s regarded, t becomes ambguous f there s stll a surface effect (Secton 2.1.7).

45 29 Bascs Consderng the fundamental equaton of Gbbs (Eq. 2.26) and the total dfferental of the Euler equaton for the Internal Energy, we obtan the Gbbs-Duhem relatonshp [Prausntz 1999]. + n d = SdT VdP µ 0 (2.57) Takng nto account the potental feld of an adsorbent surface wth area A, ths fundamental equaton extends by the nterfacal tenson σ. The extended Gbbs-Duhem equaton gves: SdT VdP + Adσ + n dµ = 0 (2.58) Wth constant P and T, Eq wll gve the Gbbs adsorpton sotherm: + Ad σ n dµ = 0 (2.59) Accordng to the extremum prncple, the equlbrum state s characterzed by a mnmum of the Internal Energy U and a maxmum of Entropy S n the fundamental varables. In equlbrum state, all state varables lke U, S, V and n are constant and the system can exst n form of a homogenous or several coexstng phases. Other equlbrum condtons are: constant temperature T, constant pressure P and equalty of the chemcal potental µ n the dfferent phases of the system (see Secton 2.2). µ can be expressed n terms of the chemcal potental of the pure compound µ 0 at system pressure and temperature, the mole fracton of the compound x n the correspondng phase and ts actvty coeffcent γ. µ = µ ( T, P) + RTln x + RTln γ (2.60) 0 Equaton 2.60 descrbes the chemcal potentals of the component n the bulk soluton whle the followng lnes mean to derve the chemcal potental n the adsorbed phase. The chemcal potental of a compound n the adsorbed phase µ ads depends on four ntensve parameters, the composton x ads, T, P and the nterfacal tenson σ. σ 0 stands for the nterfacal tenson between the pure lqud compound and the sold surface whle σ depcts the nterfacal tenson between the soluton and the adsorbent. Leavng P and T constant, the chemcal potental of the pure compound n the adsorbed layer can be derved by ntegraton of the extended Gbbs-Duhem equaton (Eq. 2.58) nstead of Eq

46 30 Bascs σ ads ads µ 0 ( σ ) µ 0 ( σ 0 ) a0 l dσ (2.61) = σ 0 Where l s the average number of adsorbed monolayers of the pure (see ndex 0 ) compound and a depcts ts adsorbed molar surface area. Wth a only dependng on T t follows: ads ads 0 σ ) µ 0 ( σ 0 ) ( σ ) µ ( = σ (2.62) a0 l 0 The term (σ-σ 0 ) depcts the free energy of mmerson nto the soluton mnus mmerson nto the pure lqud compound. The chemcal potental of the adsorbed compound can be wrtten as follows: ads ads ads ads ads ads ( γ x ) = µ ( σ ) a m ( σ σ ) RT ( γ x ) µ = µ ln (2.63) ads 0 + RT ln In analogy to the adsorbed phase and assumng constant T and P, the lqud phase chemcal potental of compound (µ lqu ) can be derved from Eq. 2.60: lqu lqu 0 + lqu lqu ( γ x ) µ = µ RT ln (2.64) In thermodynamc equlbrum, equalty of chemcal potental can be assumed: lqu lqu ads ads µ ( T, P, x ) = µ ( T, P, σ, x ) (2.65) Combnaton of Eqs. 2.63, 2.64 and 2.65 wll lead to the followng fundamental expresson for the adsorpton equlbrum between the lqud and the adsorbed phase: ( σ σ ) lqu lqu ads ads a0 0 γ x = γ x exp (2.66) m RT Descrbng adsorpton equlbrum usng adsorpton sotherms The affnty of a compound to accumulate at a phase boundary s often descrbed n terms of adsorpton sotherms, whereas the concentraton c of compound n the bulk phase s correlated to the adsorbed concentraton q. Therefore, adsorpton equlbra are determned by ther sotherms. Ths approach leads to a dfferent way n descrbng adsorpton equlbra and assocated expermental data than the one shown n the secton before (see Eq. 2.66). In 1879, a general appled thermodynamc concept to descrbe adsorpton equlbra for gas phase adsorpton was developed by Gbbs [Gbbs 1928]. Later, Langmur [Langmur 1916] and Brunauer et al. [Brunauer 1938] provded further adsorpton theores. These theores,

47 31 Bascs along wth ther mathematcal equatons represent theoretcal gudelnes to nterpret expermental adsorpton data. In the followng, a bref overvew on the feld of adsorpton sotherms s gven. All adsorpton sotherms dscussed wthn ths secton are commonly referred to as loadng sotherm. More detaled nformaton on the topc can be found n lterature [Guochon 2002]; [Ruthven 1984]. In chromatography, an adsorpton sotherm s defned as Isotherm descrbng adsorpton of the sample component on the surface of the statonary phase from the moble phase [IUPAC Gold Book 1997]. The smplest form of an adsorpton sotherm s a lnear type (type I). At very low solute concentratons c, a constant slope wll n most cases well depct the adsorpton equlbrum. It can be expressed by the Henry equaton, where k H s the so-called Henry constant. q = k c (2.67) H At hgher concentratons the concentraton overload leads to non-lnear adsorpton behavour as the number of adsorpton stes becomes restrcted. The most prevalent non-lnear adsorpton equlbrum relaton s the Langmur sotherm [Langmur 1916], whch accounts for the effects of solute nteractons and sorbent saturaton. q = q sat b c 1+ b c (2.68) Wth the Langmur factor b defned as k H b = sat (2.69) q Conformable to the Langmur approach, the loadablty q sat depcts the lmtng concentraton of solute molecules n the statonary phase. At ths concentraton, all possble adsorpton stes are occuped. In Fgure 2.7 schematcally depcts the Langmur and the type I sotherms, respectvely.

48 32 Bascs Fgure 2.7: The Langmur sotherm [Astrath 2007a] Further relatons descrbng sngle component adsorpton equlbra are: () the Freundlch sotherm beng an emprcal two parameter equaton wthout thermodynamc dervaton. All potental functons can be gathered under ths synonym. () the BET Isotherm as an extenson of the Langmur equaton to mult layer adsorpton by the assumpton of a composed adsorpton enthalpy. h = h + h (2.70) ads bondng evaporaton For compettve adsorpton equlbra of two or more components, models have been developed to calculate mult component sotherms from sngle component sotherms or on emprcal bass. Extensons of the above mentoned sotherm models are e.g. the Mult- Langmur equaton for n compounds and mult-freundlch equatons. In addton to these extensons, further models have been developed to descrbe compettve adsorpton equlbra, such as the Ideal Adsorbed Soluton Theory (IAST), Real Adsorbed Soluton theory (RAST) or the Vacancy Soluton Model (VSM). Detaled descrpton of these models les not n the focus of ths work. A phenomenologcal classfcaton of dfferent forms of loadng sotherms can be found n lterature [Kast 1988].

49 33 Bascs Lqud-lqud (absorpton) equlbrum Due to ts thermodynamc bass, the phase equlbrum approach (Secton 2.2.3) also enables the descrpton of a lqud-lqud phase equlbrum. Assumng a system of two lqud phases s n equlbrum and referrng to the same reference state of the pure compounds, Eq wll lead to: x γ = x γ (2.71) α α β β Referrng to Eq. 2.45, parttonng between two lqud phases can therefore be expressed n terms of actvty coeffcents. K αβ γ = (2.72) γ β α To provde a constant K, concentraton dependency of the actvty coeffcent has to be elmnated by assumng nfnte dluton of compound wthn both phases. That s, component s exclusvely nteractng wth surroundng solvent molecules. Therefore, no solute-solute nteractons wll occur. If so, K αβ can be expressed n terms of actvty coeffcents for nfnte dluton. x γ γ = = = (2.73) α β β αβ lm K lm lm α β α β β α β α α x, x 0 x, x 0 x x, x 0 γ γ Underlyng dstrbuton mechansm: adsorpton or partton Although elute separaton can be smply expressed n terms of dfferences n elute dstrbuton between the two auxlary phases, t s a complex thermodynamc mbalance that causes ths phenomenon and much controversy has been gong on about the molecular mechansm behnd ths mbalance. To descrbe retenton and ts underlyng molecular mechansm n lqud-sold chromatography, two lmtng models have been brought forward. In the 1960 s, Snyder [Snyder 1968] proposed the so-called dsplacement model. Wthn ths model, the elute s assumed to be dstrbuted between a sold surface (statonary phase) and a moble phase as a result of a compettve solute and solvent adsorpton. The other lmtng model wll not assume adsorpton onto a surface but the transfer of a solute from one bulk soluton to another mmscble solvent, qute analogous to the elute parttonng n a lqud-lqud chromatographc system [Dll 1987]; [Dorsey 1989].

50 34 Bascs The queston f, n lqud-sold chromatography, retenton s governed by an adsorpton or a partton (absorpton) mechansm has not yet been resolved and s stll object to ongong research. Ths tedous process reflects the complexty of underlyng mcroscopc mechansms and the lmtaton of expermental means to gan nsght Absorpton as underlyng molecular separaton prncple Chromatographc retenton can also be trggered by a partton lke mechansm between two bulk phases, as beng the case n the so-called lqud-lqud chromatography (LLC). Regardng a bonded phase wth fur-lke alkyl chans on sold partcles, the nteractng part of the statonary phase can be vewed as some knd of bulk phase, leadng to absorpton as underlyng prncple and therefore beng conceved as partton mechansm. 2.3 Modelng of chromatography The modelng of chromatographc processes ams to tackle dvers objectves n the context of elute separaton, as there are () the dentfcaton of deal separaton condton relatve to a gven elute mxture, () the predcton of system specfc elute behavour and () the dentfcaton of resultng chromatographc peaks. As can be seen from Secton 2.1 and sectons theren, the chromatographc separaton process s complex and underlyng mechansms are not yet fully resolved. Therefore, modellng such a process s a rather dffcult task. So far, bascally two groups of approaches have been used to model chromatography: () one s based on theoretcal understandng of physochemcal processes, leadng to fundamental thermodynamc and mass-transfer relatonshps (see Sectons and 2.3.2). () another popular way s based on large emprc coeffcent databases obtaned prmarly from expermental measurements that are then used to extrapolate towards unknown systems (see Secton 2.3.3) Mass balance based models of chromatography Mass balance approaches have been developed to lnk the mechansm of elute behavour n a column to ts band profles. Bascally, these methods am for solvng mass transfer knetcs and dfferental mass balance equatons n a defned slce of a chromatographc column. In combnaton wth proper nput parameters from e.g. sotherms and knetc measurements, these mathematcal approaches have good predctve potental [Felnger 2004]; [Qunones 2000].

51 35 Bascs Frst mass balance equatons dealng wth chromatographc applcatons were derved n the frst half of the last century [Bohart 1920]; [Wcke 1939a]; [Wcke 1939b]; [Weyde 1940]. Assumng a partton mechansm (Secton 2.2.6), Wlson [Wlson 1940] was frst to apply dfferental mass balance equaton to a chromatographc process. Later, Wang et al. [Wang 1978] used t to descrbe retenton n lqud chromatography. As the establshment of a mass balance model les not wthn the am on ths work, the gven nformaton remarks shall satsfy the requrements. For further nformaton on ths subject see e.g. Golshan-Shraz and Guochon [Dond 1992] who have extensvely revewed and categorzed mass balance approaches Actvty coeffcents models Regardng lqud-lqud chromatographc applcatons, knowledge about the system phase equlbrum (Secton 2.2.5) s essental to predct retenton. In lterature evdence exsts, that for some reversed statonary phase partcles, the bonded phase can be regarded as lqud-lke (see Secton ). In that case, the lqud-lqud equlbrum approach can be extended onto the vast feld of RP-HPLC. To predct the elute k -factor, t s necessary to know the column phase rato Φ (Eq. 2.55) and the partton coeffcent K αβ (Eq. 2.56). K αβ n turn can be calculated from Eq. 2.72, whch requres the elute actvty coeffcents n both lqud phases. In thermodynamcs, the actvty coeffcent descrbes the devaton of a real mxture from a predefned deal mxture, makng both mxture propertes comparable. In the followng, a short overvew wll be gven on actvty coeffcent models that have been used n context wth equlbrum predcton n chromatographc applcatons. Relevant models are: () the NRTL equaton [Chen 2005], () the UNIFAC equaton [L 2003] and () the COSMO-RS model whch has been used to predct lqud-lqud equlbrum n counter current chromatography [Hopmann 2011] The NRTL equaton Ths equaton called Non-Random Two Lquds was developed by Renon and Prausntz [Renon 1968] wth the assumpton of an nteracton between a centre molecule and ts drect neghbours. A non-arbtrary arrangement around the centre molecule s assumed, makng a thrd emprcal parameter necessary. From bnary mxture parameter data t s possble to calculate actvty coeffcents for multnary mxtures. Parameters of about 1000 LLE and over VLE bnary mxtures can be found n the Dechema Chemstry Data Seres.

52 36 Bascs The UNIQUAC equaton Accordng to ts defnton, the actvty coeffcent can be dvded nto two contrbutons: An () entropc or combnatoral contrbuton and a () resdual term, holdng the enthalpc contrbuton. ln γ = lnγ + lnγ (2.74) comb res In contrast to the NRTL equaton, n whch local volume fractons referrng to the whole molecule are beng used, Abrams and Prausntz [Abrams 1975] break down the molecule descrpton nto segmental area and volume contrbutons. Ths results nto the mplementaton of parameters that account for the numbers of segments per molecule and the relatve molecule surface and volume, whch Abrams frst found n the glas book of Bond. Values for these parameters can be found n lterature (Dechema Chemstry Data Seres, Volume I). Another temperature dependent energy parameter has to be adopted by expermental data and can partly be found n lterature The UNIFAC equaton In 1925 Langmur was frst to suggest the estmaton of thermodynamc propertes of lqud mxtures from group contrbutons. Based on ths dea and the UNIQUAC model [Abrams 1975], Fredenslund et al. [Fredenslund 1975]; [Fredenslund 1989] developed the Unversal Functonal-Group Actvty Coeffcent method n The basc dea behnd the UNIFAC method s to use exstng phase equlbrum data for predctng phase equlbra of other systems. It s an ncremental method, dvdng molecules nto structural groups and therefore reducng the vast amount of chemcal compounds to about 100 dfferent structural groups. If parameters of all structural group nteracton have been ftted to experment, t becomes theoretcally feasble to predct phase equlbra of any possble system. The UNIFAC equaton (UNIFAC-Dortmund) for the temperature ndependent combnatoral contrbuton calculates as follows: ' ' comb ϕ ϕ 1 θ ϕ ln γ = ln 1 zq ln 1 x x 2 ϕ θ (2.75) wth θ and φ beng the molar weghted segments and fractonal components of compound respectvely.

53 37 Bascs θ = ' ϕ = j x q j x j x r q j x jr (2.76) (2.77) The resdual term wrtes: ( ) ( ln Γ ln Γ ) res ( ) lnγ = υ k k k (2.78) k wth Γ k () beng the segment actvty resultng only from nteractons wth compound. Θ mψkm m ln Γk = Qk 1 ln Θ mτ mk (2.79) m Θ nψnm n Here, Θ m s the summaton of the surface area fracton of segment m over all other segments. The nteracton parameter Ψ mn s calculated as follows: Ψ mn = exp ( U U ) mn RT nn a = exp T mn (2.80) To calculate Θ m, the segment mole fracton X n has to be determned frst. Θ = Q n m Q X n m X X m = ( j) ν x j j j ν ( j) m n x n j m (2.81) (2.82) = k r ν R = k ( ) k q ν Q ( ) k k k (2.83) (2.84)

54 38 Bascs In contrast to the UNIQUAC equaton, r wll be expressed by the number v () k of structural groups k n compound, wth R k beng a volume parameter for the structural group k. q s calculated usng the reduced surface area Q k of group k. Both parameters r and q are retreved by fttng to expermental values and can be found n lterature (e.g. Dechema Chemstry Data Seres). Today, the parameter matrx of the 100*100 structural groups s already flled to a hgh percentage The COSMO-RS model The Conductor-lke Screenng Model for Real Solvents (COSMO-RS) by Klamt [Klamt 1995] permts an a-pror predcton of thermodynamc propertes, based on molecular structure only. By calculatng the chemcal potental of a molecule n an arbtrary conglomerate of other substances, COSMO-RS s capable to compute e.g. the actvty coeffcent γ of a compound n a mxture. Ths route of predctng data was ntroduced by one of the authors n 1995 [Maassen 1995]. For more detaled nformaton see Secton Exothermodynamc models Another way to predct parameters n a chromatographc system are statstcally derved relatonshps between molecule descrptors (emprcal or theoretcal parameters derved from physochemcal characterstcs of a molecule) and measurable molecule propertes. As these emprcally derved correlatons lack the rgor of thermodynamcs, they are often referred to as exo- or extrathermodynamc relatonshps. In contrast to thermodynamc approaches that wll merely lead to system bulk propertes, exothermodynamc relatonshps can offer the chance to provde nsght to underlyng physochemcal phenomena. Chromatography offers a great feld of applcaton for dvers PPAs (Property predcton approaches). They have been appled extensvely n chromatography and a great number of publcatons addressng ths topc can be found. Dfferent model mplementatons exst wth partally overlappng defntons. Therefore, a certan applcaton mght ft nto several model defntons; e.g. a retenton tme predcton relatonshp usng molecular elute descrptors could lkewse be labeled QSRR, LFER or LSER model. Despte ths ambguty, the followng sectons am to gve short revew on the topc. Furthermore, the author has undertaken an attempt to condense the wrtten nto a more lucd sketch (Fg. 4.27).

55 39 Bascs QSRR For a set of structurally dverse elutes, quanttatve structure-retenton relatonshps (QSRR) can be derved n terms of statstcal relatonshps between parameters from chromatographc measurements (retenton data) and numercal measures (descrptors) of elute-, moble and statonary phase propertes. Examples for commonly used chromatographc parameters are retenton tme t R, capacty factor k or k W (capacty factor wth pure aqueous moble phases). Molecular descrptors n turn can be obtaned by standardzed experments (expermental descrptors) or molecular characterstca are used to comple theoretcal descrptors. For the great spectrum of elute descrptors see extensve tabulatons of the followng revew paper [Heberger 2007]. A great number of QSRR studes exst, dealng wth dfferent aspects of chromatography. Nevertheless, the fundamental objectve of QSRR s to predct elute retenton by means of molecular characterstcs (descrptors) as ndependent parameter. For more detaled nformaton on QSRR see recent revews [Kalszan 2007]; [Heberger 2007]; [Put 2007] QSAR Hour of brth of modern QSAR (Quanttatve Structure-Actvty Relatonshp) methods was n 1964 by work of Hansch et al. [Hansch 1964]. Later, the dea to statstcally relate the bologcal actvty of a compound to ts structural characterstcs was appled to the analyss of chromatographc data (see QSRR). Snce then, quanttatve structure-actvty relatonshps have been employed to predct propertes lke boavalablty [Agrawal 2002]; [Yang 1996]; [Yoshda 2000], blood-bran parttonng [Wchmann 2007]; [Crvor 2000], toxcty [Clark 2002]; [Cronn 2000] or skn permeablty [Cronn 1999]; [Gute 1999] QSPR The term Quanttatve structure-property relatonshp s often used to name relatonshps between solute descrptors and physochemcal propertes n general. QSPRs and QSARs have smlar methodology; hence, n lterature they are often used alke. Conceptually QSAR/QSPR also resemble the QSRR approach. Whle the latter ams for retenton data predcton, QSAR/QSPR wll use ths data as an ndependent varable to estmate molecule propertes [Heberger 2007]. For yet uncharacterzed compounds, QSPR s frequently appled to predct ADME (absorpton, dstrbuton, metabolsm, and excreton) propertes [Ramamurth 2004]; [Obrezanova 2008].

56 40 Bascs LFER Hammet equaton [Hammett 1937] and Brønsted relaton are beng regarded as famous examples for so-called Lnear Free-Energy Relatons. These early LFERs showed the possblty of correlatng molecular structure parameters to measurable emprcal propertes (e.g. dstrbuton coeffcent, rate constant) by usng parametrc procedures. A change of name to Lnear Gbbs Energy Relaton was suggested by IUPAC allowng for the fact that LFERs express the proportonalty of a molecular structure change to a standard Gbbs energy change. G 0 n = j 0 G (2.85) wth G 0 beng the elementary Gbbs energy change connected to a structural fragment of a compound holdng a number j of smlar fragments. G 0 s n turn thermodynamcally lnked to measurable propertes lke the partton coeffcent K (Eq. 2.51). In 1949, the LFER concept was ntroduces to chromatography by A. Martn [Martn 1949]. Regardng Gbbs energy related retenton data, LFERs show the ablty to ndependently grasp ndvdual molecular nteracton contrbutons to the retenton process. For that purpose, descrptors that resemble Gbbs energy related quanttes are necessary. Abraham and coworkers ntroduced sutable parameters whch were derved from molecule structure or equlbrum measurements [Abraham 1993] [Abraham 1994]. Retenton data, such as the commonly used capacty factor k can be correlated to those descrptors usng the LFER or solvaton equaton ntroduced by Abraham et al. [Abraham 1993]: H H α 2 + b β + v H log SP = c + r R2 + s π 2 + a 2 Vx (2.86) SP stands for any elute parttonng related measure (e.g. log k ), c s the ntercept, R 2 s an excess molar refracton, π H 2 s the elute dpolarty/polarzablty, Σα H 2 and Σβ H 2 are the elute overall or effectve hydrogen-bond acdty and bascty, and V x s the McGowan characterstc volume. In chromatographc applcatons, where elute parttonng between two phases s nvestgated, the LFER coeffcents (r, s, a, b and v) wll express the dfferences between the phases. LFER coeffcents can be calculated by multvarate regresson analyss. In ths context, LFER provdes a frm theoretcal bass to predomnantly emprcal relatonshps as QSAR, QSRR or QSPR. Durng the last decades, varous forms of LFERs have found wde currency n chromatography. One of the most wdely known s the CLOGP method of Leo and Hansch [Hansch 1979] for octanol-water partton coeffcent (P OW ) predcton, where CLOGP descrptors are usually calculated from molecular structure [Kalszan 1999];

57 41 Bascs [Kalszan 1999]. In general, P OW values can be calculated accordng to Leo and Hansch as follows: f + log = F (2.87) P OW Where f stands for the fragmental contrbutons and F s the so-called fragmental factor, accountng for the fragmental poston wthn the molecule. The CLOGP parameter also fnds applcaton n several QSAR studes [Colmenarejo 2001]; [Gute 1999]; [Clark 1999] or n QSRRs, e.g. the smple model of Martn and Synge [Martn 1941], relatng elute partton coeffcent to ts retenton LSER Both models, LFERs as well as Lnear Solvaton Energy Relatonshps are based on the same methodology. Addtonally, LSER nput parameters and notaton have not been statc over the years, makng dfferentaton of the two models not an easy task. An ntersecton of both models can be seen, where solublty related propertes are beng descrbed usng Gbbs energy related parameters (descrptors). As far as parttonng and therefore solublty related propertes from chromatographc measurements are concerned, all LFERs can hence be vewed as LSERs. Over the last decades, LSER focus has changed from descrpton of bulk solvent to elute behavour but the basc prncple behnd the model has remaned the same. Elute transfer s stll regarded as a three step process: () cavty formaton wthn bulk solvent, () elute transfer nto the cavty and () closng of the cavty left behnd. Poneerng work towards LSER development n chromatography was done by Kamlet, Taft and coworkers [Kamlet 1976a]; [Kamlet 1983]; [Kamlet 1988], who used e.g. descrptors from UV-vsble shft experments to elucdate molecule characterstcs nvolved n retenton and other elute parttonng related propertes. Due to the descrptors derved from spectroscopc measurements, ths approach has often been referred to as solvatochromc comparson method. More detaled nformaton concernng LSERs can be found n the followng revew paper [Vtha 2006]. 2.4 Chemcal and quantum-chemcal bascs The term quantum chemstry outlnes the applcaton of quantum mechancs and quantum feld theory to problems n chemstry. It ams for a descrpton of matter on a molecular scale. Quantum mechancs bases on a seres of postulatons. The most mportant one mples that the

58 42 Bascs complete state nformaton of a physcal system s held by a wavefuncton Ψ (r1,r2,...,rn, t), where r depcts the poston vectors of n system partcles whle t stands for the tme. The squared wavefuncton 2 Ψ depcts the probablty of atom presence n space [Preuß 1990]; [Preuss 1972]. Usng quantum chemstry applcatons, molecule geometres and conformatons can be descrbed by nteractons of electrons and atomc nucle. Ths approach bases upon the Schrödnger equaton whch delvers a mathematcal descrpton of the dfferent states of electrons that move around the atomc nucleus. Snce the Schrödnger equaton s not to be solved on analytcal bass, methods exst that am to approxmate a soluton as can be seen n the next Secton Ab-nto and sem emprcal methods Dependng on the type of approxmate soluton to the tme-ndependent Schrödnger equaton, dstncton can be made between ab-nto and sememprcal methods. Whle sem emprcal methods mply expermental nformaton to the solvaton procedure, ab-nto methods merely use values of the fundamental physcal constants and nuclear charges of the atoms concerned. The so-called Hartree-Fock (HF) approxmaton (1930) can be consdered as an example of an ab-nto method. Here, the Schrödnger equaton s solved teratvely wth the ad of approxmated wavefunctons. Lookng at an arbtrary electron, only nteractons between the sngle electron and the surroundng space-charge cloud wll be consdered. The charge cloud n turn s formed by tme averaged postve core and negatve electron charges. Therefore, surroundng charge moton wll have no nfluence on the movement of the regarded electron. Hence, a mult electron wavefuncton can be splt nto sngle electron wave functons, also called spn orbtals. These sngle lnear ndependent functons are denoted as bass set. To correctly represent the ntegrty of molecule orbtals, an nfnte number of such functons s necessary. To stay wthn the realms of possblty, an approxmate wavefuncton can already be realzed by a lmted number of functons. Despte of the descrbed smplfcatons, applcaton of HF approxmaton stll means a lot of computng tme. Based on the same conceptual framework as ab-nto methods, sem emprcal methods are capable to drastcally reduce computng tme by ntroducng addtonal model smplfcatons. One mportant approach s to reduce the amount of electrons regarded to the exclusve consderaton of valence electrons. Another key approach wll lead to neglgence of dfferental overlappng. On the other hand, errors wll be caused by addtonal approxmatons. Therefore, remanng ntegrals wll be replenshed wth emprcal parameters

59 43 Bascs calbrated aganst expermental or theoretcal reference data, such as molecule geometres and enthalpy of formaton. Sem emprcal methods prove to be effectve computatonal tools, yeldng fast quanttatve estmates for a number of propertes. MNDO [Dewar 1977], AM1 [Dewar 1985] and PM3 [Dymek, Jr. 1989] are the most wdely known representatve for ths group of methods. For more detaled nformaton see the followng lterature [Scholz 1981]; [Clark 1985] or revews [Stewart 1990]; [Thel 1996] on the subject Densty functonal theory Densty functonal theory (DFT) descrbes the energy of an atom or molecule by usng the energy functonal E(ρ e (r)), whch only depends on ts electron densty ρ e (r). Compared to 3n dmensonal wavefuncton-based ab-nto or sem emprcal methods, DFT provdes the clear advantage of the electron densty only beng a functon n 3 dmensonal space. Usng an dealsed model of a unform electron gas, Thomas und Ferm (1927) were frst to employ electron densty for descrpton of atoms and molecules. Usng dealzed assumptons, an expresson for the electron knetc energy T TF can be deduced. Combnng electrostatc ueg ueg actve core-electron E Ne and repulsve electron-electron E ee nteractons wth the expresson for the electron knetc energy wll yeld the so-called Thomas-Ferm Energy Functonal E TF [Koch 2002]. E TF ueg ueg ( ρ ( r) ) T ( ρ ( r) ) + E ( ρ ( r) ) + E ( ρ ( r) ) e = (2.88) TF e Ne e ee Totally neglectng replacement and correlaton nteractons, the Thomas-Ferm Energy Functonal can only be consdered as a very rough approxmaton of the exact theory. Nevertheless t only depends on the electron densty as the sngle varable. Bass of the DFT n current applcaton are the so-called Hohenberg-Kohn Theorems (1964) [Koch 2001]. The frst theorem justfes the applcaton of the electron densty as the fundamental DFT varable. The fact that the energy of a quantum chemcal system can be expressed through the energy densty functonal has been proven by Hohenberg and Kohn. System energy can hence be calculated from atom or molecule ground state electron densty ρ 0, whch s characterzed by a mnmum n total energy E 0. The electron densty ρ n turn determnes the knetc electron energy T, the electron-electron nteractons E ee and the external potental V ext. Furthermore, the electron densty defnes the ground state, the ground state wavefuncton and all atom or molecule characterstcs. The frst Hohenberg-Kohn Theorem states, that f an external potental V ext descrbes a system ground state, there can be only one V ext as a functon of a sngle electron densty ρ e (r) [Koch 2001]; [Parr 1989]. e

60 44 Bascs Due to the work of Hohenberg and Kohn, the total energy of a system ground state can be splt nto a system dependent and a unversal part. ( ρ ) E ( ρ ) + F ( ρ ) = E ( ρ ) + T ( ρ ) E ( ) E = + (2.89) 0 0 Ne 0 HK 0 Ne 0 0 ee ρ0 The expresson ( ρ ) E ( ) T + depcts the system ndependent Hohenberg-Kohn energy 0 ee ρ 0 functonal for those both terms, the knetc energy T and the electron-electron nteracton E ee, no explct expresson exsts [Koch 2001]. Correspondng to the second Hohenberg and Kohn Theorem, the ground state energy E 0 can be calculated exactly usng varatonal calculus. But ths only holds true, f ground state electron densty ρ 0 s used, as t becomes obvous n followng equaton. E ( ρ '( r) ) = E ( ρ' ( r) ) + T ( ρ' ( r) ) + E ( '( r) ) E (2.90) 0 Ne ee ρ In vew of the ndependent Hohenberg-Kohn functonal, n 1965, Kohn and Sham presented a way to approxmate ths unknown functonal [Koch 2001]. Knowng the descrpton of the knetc energy to be mpossble f usng an explct densty functonal, Kohn and Sham suggested the applcaton of a non-nteractng reference system. Here, the knetc energy of a real system wll be substtuted by the knetc energy of a non-nteractng sngle partcle system T s (ρ) and an addtonal correcton term T C. T s (ρ) can be descrbed by orbtal functonals and T C s meant to grasp knetc energy dfferences between a real and a sngle partcle system. T ( ρ ( r) ) T ( ρ( r) ) + T ( ρ( r) ) = (2.91) S C Lookng at the unknown electron-electron nteracton term E ee, a soluton procedure smlar to the one used for T can be appled by splttng E ee nto known and unknown parts. Ths procedure leads to a classcal electron nteracton term E J, and a correcton term E ncl, ntegratng all non-classcal parts of E ee. E ee ( ρ ( r) ) E ( ρ( r) ) + E ( ρ( r) ) = (2.92) J ncl Both terms T C and E ncl can be joned to a combned correcton term E XC. E XC ( ρ ( r) ) T ( ρ( r) ) + E ( ρ( r) ) = (2.93) C ncl Combnng Eq. 2.91, 2.92, 2.93 and the frst Hohenberg-Kohn Theorem, wll lead to the followng descrpton of the atom or molecule energy,

61 45 Bascs E ( ρ ( r) ) T ( ρ( r) ) + E ( ρ( r) ) + E ( ρ( r) ) + E ( ρ( r) ) = (2.94) S Ne J where T S depcts the electron knetc energy and E Ne stands for electron-nucleus nteractons. Assumng electron moton to be ndependent from each other, E J represents the electronelectron nteractons, whle E XC accounts for the exstng moton dependence between electrons wthn the nucleus charge feld. For E XC as a stll unknown term, approxmatons can be found n lterature [Koch 2001]. Due to advantages as the use of only vew unversal parameters, DFT has become a standard method for molecule geometry and electron densty calculaton. Model progressons have been made towards the calculaton of excted state systems by ntegraton of the parameter tme. Ths so-called tme dependent densty functonal theory (TDDFT) fnds applcaton n commercal quantum chemcal programs lke the one used n the present work (Turbomole) (see Secton 2.4.3). More detaled nformaton on DFT can be found n lterature [Koch 2001]; [Parr 1989]; [Drezler 1990]; [Casda 1995]. XC Contnuum solvaton models Many concepts such as the one presented n the prevous secton have been created for molecule structure and phenomena nterpretaton and analyss for systems n the gas phase. To expand usablty towards solvated systems, solvent effects have to be consdered. Wth other words, t s necessary to mplctly consder molecules that surround a solvated molecule. Ths can be acheved by a contnuum representaton, where surroundng molecules are regarded as a contnuum and molecule responses towards the contnuum are accounted for. The molecule energy dfference between gas phase and solvated phase s expressed by the Gbbs energy of solvaton G solv, accountng for the Gbbs energy change of a molecule mgratng from gas phase to condensed phase. Free energy change conssts of two parts: () placement of a partcle nto a fxed poston of the system and () elmnaton of the fxed poston restrants. Ths soluton process defnton and the resultng system energy change have become prevalent n solvaton thermodynamcs [Tomas 1994]. The solvaton process chemcal potental of a component can be descrbed as follows: * 3 = µ + kt ln ρλ µ (2.95) µ * depcts the pseudo chemcal potental of component wth all partcles fxed at ther postons, whle kt ln ρ Λ accounts for the dfference n chemcal potental between fxed 3

62 46 Bascs and free partcles. ρ denotes the partcle number densty of component and Λ 3 comprses ntegraton over all translatonal moments of a molecule. T and k stand for the absolute temperature and the Boltzmann constant, respectvely. Correspondng to the salvaton process defned above, the change n free energy G solv can be descrbed as follows: sol cav rep dsp el G = G + G + G + G (2.96) The four terms descrbed n the equaton above wll have the followng meanng. G cav stands for the free energy fracton, necessary to form a cavty wthn the bulk soluton ft to ncorporate the solvate molecule and an addtonal entropc part resultng from solvens molecule shft around the cavty. G rep and G dsp represent the part of free energy change arsng from repulsve and undrected dspersve nteractons, respectvely. The last term G el depcts the free energy change from electrostatc nteracton. An addtonal term ( G hb ) can be added to explctly descrbe the free energy change contrbuton of hydrogen bonds nteractons [Leach 2001]. Contnuum salvaton models explctly descrbe the electrostatc nteracton term G el, wth the contnuum beng a homogenous, sotropc electrcal feld. Ths feld can be characterzed by a sngle scalar varable, the so-called delectrc constant ε. Basc prncples of ths class of contnuum models orgnate from the work of Born [Born 1920], Onsager [Onsager 1936] and Krkwood [Krkwood 1939]. Contnuum solvaton models defne a cavty, where the solvated partcle s placed. Determnaton of cavty shape and sze s a crucal task. If the cavty s chosen to small, atom electron denstes close to the cavty border can be mscalculated. An overestmaton n cavty sze on the other hand may lead to an attenuaton of the contnuum effect. Lookng at the solvated partcle, charge densty and dstrbuton are essental parameters when t comes to partcle-solvent nteracton determnaton. The partcle charge densty can be calculated by quantum mechancal means. Wth defned cavty and known partcle volume charges, the concept of the self-consstent reacton feld (SCRF) can be appled to teratvely attan analogy between cavty and solvate molecule geometry and charge dstrbuton. The SCRF approach wll be explaned later on, usng the smplfcaton of surface charges. Lookng back at the problem of cavty determnaton, t can be seen that popular contnuum models lke the Polarzed Contnuum Model (PCM) [Mertus 1981] or the Conductor-lke Screenng Model (COSMO) [Klamt 1993] use overlappng spheres to handle the challenge. Thereby, spheres of a radus approxmately 20 % bgger than the van-der-waals radus wll be appled around the atoms of the solvate molecule, resultng nto cavty formaton. Yet another

63 47 Bascs smplfcaton s appled wthn the mentoned models. Volume charge denstes wthn the 3- dmensonal mesh wll be projected onto a surface, yeldng surface charge denstes. Ths procedure wll drastcally reduce the electrostatc complexty and therefore smplfy SCRF applcaton as explaned n the followng. Frst, an electrostatc feld of the solvate molecule n vacuum wll be calculated from ts surface segment pont charges. Ths electrostatc feld s then projected to the cavty surface, leadng to the screenng charge densty. Screenng charges for ther part apply a reacton feld onto the solvate molecule, leadng to geometry and charge dstrbuton changes f consdered wthn the functonal of DFT calculatons (Secton 2.4.2). Ths crcutry of geometry and charge densty changes wll lead to an teratve process that s contnued untl self consstency s reached. At that pont, the electrostatc feld and surroundng cavty screenng charge densty wll concde. For detaled nformaton on SCRF models see followng lterature [Cramer 1995]; [Cramer 1999]. Charge denstes of the cavty surface can be calculated by the followng equaton known from electrostatcs [Leach 2001]; [Klamt 1998]. 4πσ ε 1 = (2.97) ε ( r) n( r) E( r) Surface pont r charge densty σ(r) s dependent on the delectrc constant ε, the normal vector n(r) and the force of the electrc feld E(r). By assumng the contnuum to be a perfect conductor and consequently settng the delectrc constant to nfnty (ε=0), the COSMO model by Klamt and Schüürmann can be consdered as a partcularly effcent type of contnuum solvaton model. Further mathematcal smplfcaton was acheved by dvdng the solvent accessble molecule surface nto a fnte number of segments of constant charge densty σ. Every segment s also explctly assgned to an atom. All these smplfcaton lead to the possblty of drect contnuum ntegraton nto the DFT energy functonal. Therefore molecule geometry and charge densty dstrbuton can be calculated on a hgh quantum mechancal level. The scalng of screenng charges allows for dfferences between an deal conductor and the real solvent and s based on a scalng functon by Krkwood [Krkwood 1939]. σ ε 1 σ ε * = (2.98) σ and σ* depct the screenng charges of a real and an deal conductor, respectvely.

64 48 Bascs In presence of the screenng solvent, COSMO calculatons lead to the self-consstent geometry of a molecular structure, the delectrc screenng energy, the molecular surface A and the surface segment charge denstes σ. The energy dfference of a solvate molecule between ts resdence n vacuum and n an deal conductor s expressed by the delectrc screenng energy, yeldng the extent of electrostatc nteracton durng solvaton procedure The Conductor-lke Screenng Model (COSMO-RS) The COSMO-RS model by Klamt [Klamt 1995] s an extenson of the COSMO model and permts an a-pror calculaton of the chemcal potental of a compound wthn an arbtrary number of other components or ther mxtures. COSMO-RS vews molecule nteractons as par wse nteractons of surface segments (Fg. 4.8). Fgure 2.8: Molecule nteracton, pctured as ensemble of parwse nteracton surface segments. A: deal screenng between two segments. B: msft. C: hydrogen bonds; [Klamt 2000] Surface segment nteractons from COSMO calculatons are ntegrated nto a self consstent expresson for the chemcal potental, derved from thermodynamcs. Ths procedure allows for actvty coeffcent calculaton of a component. Every molecule surface segment s assocated wth an averaged and constant screenng charge densty σ, whch s determned by DFT calculatons usng COSMO boundary condtons (Secton 2.4.3). It should be mentoned, that screenng charges depct the reverse of the molecule partal charges by screenng them from vcnty. The surface segments are agan determned by an expermentally ftted radus r av. From the probablty dstrbuton of these segment screenng charges p(σ), a charge dstrbuton functon, the so-called σ-profle can be establshed for each molecule. By ths means, every component can be dstnctvely defned. Ths step of σ-profle establshment

65 49 Bascs transfers ths model to the advantageous and dsadvantageous of statstcal thermodynamcs. Fg. 2.9 llustrates the man procedure steps leadng to the σ-profle of water. Fgure 2.9: Development of a charge densty dstrbuton (σ-profle) from molecular structure Component mxtures n turn can be represented by weghtng the pure component dstrbuton functons p (σ) accordng to molar mxture composton. p = ( ) x p ( σ ) mx σ (2.99) Dssolved wthn an deal conductor and perceved as ensemble of surface segments, the chemcal potental µ of a component can be derved. Usng the mxture σ-profle from Eq. 2.99, the σ-potental can be calculated teratvely, usng the followng mplct equaton. RT A ' eff ' ' ' ( σ ) ln p ( σ ) exp ( µ ( σ ) E ( σ, σ ) σ µ S = S S msft d A (2.100) eff RT µ S (σ) depcts the affnty of a system S towards segments carryng a charge densty σ and A eff stands for surface segment areas that are effectvely brought n contact wth each other. A eff n turn s determned by the once expermentally ftted segment radus r eff. Wth known A and A eff, the number of surface segments n = A Aeff can be calculated. E msft arses from a nondeal parng of surface segments as descrbed n Fg. 2.8 and can be calculated as follows:

66 50 Bascs ' ' α ' ( σ, σ ) A e = A ( σ + σ ) 2 E = (2.101) msft eff msft eff 2 Wth the msft energy factor α beng a global, expermentally ftted constant, meant to correct the solvent approxmaton (Eq. 2.98). In general, the chemcal potental of component wthn a system S can be calculated usng the followng equaton [Spuhl 2006]: µ = + + (2.102) el comb, S δa µ, S µ, S Eq explctly accounts for the surface proportonal and electrostatc share of the Gbbs energy of solvaton. The delectrc screenng energy, and the molecular surface A can be determned from COSMO calculatons. The dsperson constant δ of a molecule can be calculated from the sum δ k Ak of element specfc constants δ k and correspondng surface parts A k, wth k beng the number of dfferent elements wthn the molecule. The electrostatc porton µ,s el of the component chemcal potental can be obtaned by ntegratng the product of component σ-profle p (σ) and system σ-potental µ S (σ). ( σ ) = p ( σ ) µ ( σ ) σ el µ, S S d (2.103) The electrostatc share of hydrogen bonds s well descrbed by the COSMO-RS model. However, addtonal energy, resultng from mutual electron shell penetraton of hydrogen bond donator and receptor s not suffcently accounted for. To overcome ths shortfall, the nteracton energy E, so far merely dependng on msft Energy E msft, wll by supplemented by an addtonal term: E hb ' ( σ, σ ) = A e = A c mn{ 0,max( 0, σ σ ) mn( 0, σ + σ )} eff hb eff hb acc hb dom hb (2.104) The threshold values σ hb and c hb depct once expermentally adjusted, global constants. Formulaton of Eq leads to ts consderaton only f the amount of strongly postve or negatve charge denstes of nteractng components exceeds the σ hb value. Therefore, Eq represents a threshold term. The combnatoral share µ comb,s of the component chemcal potental can be obtaned by followng equaton [COSMOtherm 2008]:

67 51 Bascs comb r q µ, S = RT λ0 ln r + λ1 1 ln r + λ2 1 ln q (2.105) r q λ 0, λ 1 and λ 2 are parameters, whch have been ftted to expermental phase equlbra data by Klamt [Klamt 1998]. The combnatoral part of the chemcal potental s represented by the Staver approach, already used n the UNIQUAC / UNIFAC model (Secton 2.3.2) COSMO calculated σ-moments Instead of usng the mplct Eq for the calculaton of the chemcal potental, µ S (σ) can also be put nto a Taylor seres-lke expresson. Wth S m ( σ ) c f ( σ ) µ (2.106) = 2 S and f ( ) σ σ = for 0 (2.107) ( σ ) ( σ ) f 2 / 1 = f acc / don (2.108) Ths leads to Eq where the partton coeffcent as well as the capacty factor of e.g. an elute between moble and statonary phase can be expressed as a lnear combnaton of ts σ- moments. m SP, MP SP, MP X log k' = c + c M (2.109) = 2 The coeffcents c hold the system nformaton of the auxlary phases whle the σ-moments M X are defned by ( σ ) f ( σ ) X X M = p dσ (2.110) p X (σ) depcts composton functon of the molecular surface wth respect to the polarzaton charge densty σ. Therefore, t s possble to calculate the σ-moments of a molecule from ts mere structure and ts deduced σ-profle. These σ-moments can be understood as structural descrptors that can be generated from the structural propertes of the molecule as ndependent

68 52 Bascs varables. Beyond the mathematcal analyss, dfferent physcal propertes can be assgned to the σ-moments: The zeroth moment depcts the molecular surface. The frst moment can be left out for neutral components because t s the total charge of the molecule. The second moment represents the molecule total polarty, whle the thrd moment llustrates the σ-profle asymmetry. The hydrogen bondng moments are a measure for the H-acceptor and the H-donor propertes of the molecule. These pure theoretcally deduced moments show great smlarty to the 5 LSER descrptors of Abraham et al. [Zssmos 2002] and therefore provde the opportunty to descrbe molecular nteractons by usng Quanttatve Structure Property Relatonshps (QSPR) [Klamt 2001]. The fundamental dfference between the σ-moments based QSPR approach and the COSMO- RS model s that the former lacks the statstcal thermodynamc step that leads from gven σ- profles to the molecule chemcal potental and fnally to the actvty coeffcent (see Secton above). QSPR model regresson requres expermental data (tranng data) that can be used to calculate the system specfc coeffcents c. Wthn ths work, multple lnear regresson (MLP) was used to estmate these dependent varables. The use of such varables also depcts the major draw back of the QSPR methods. σ-moments are by defnton not capable to descrbe the separaton system specfc propertes of the moble and statonary phase. Therefore, QSPR coeffcents need to be adopted va expermental data regresson to each new separaton system. In contrast to COSMO-RS a-pror partton coeffcent predcton, QSP relatonshps are dffcult to be extrapolated towards new separaton tasks. Due to ths reason, the σ-moment approach s only margnally nvestgated wthn ths work. As to the denomnaton, t must be mentoned that f elute retenton s nvestgated as property, QSPR wll be named QSRR (Qanttatve Structure Retenton Relatonshp) Conformatonal analyss Wth regard to the COSMO-RS model, bass of actvty and partton coeffcent predcton are the molecular structures of all components wthn a consdered mxture. However, molecule structures cannot merely be vewed as rgd or nflexble, but rather can they form dfferent conformatons. Due to the temperature dependent Brownan moton, all molecules are constantly stmulated. As a result, molecules wll mantan a certan three dmensonal structure only for a very short tme span (e.g. ethan for approxmately s). The resdence tme of a structure wll ncrease wth decreasng potental energy. Lookng at a great number of conformatons, the relatve porton of a certan structure wthn ths mxture of

69 53 Bascs conformatons s proportonal to ts relatve resdence tme. Usng statstc mechancal means, t can be shown, that the probablty of occurrence of a certan conformatonal structure s representable by a Boltzmann dstrbuton respectng the potental structure energy [Leach 2001]. An ndrect confrmaton of ths approach was accomplshed by realstc modellng of expermental onzaton and NMR spectra [Nkk 2001]. Dependng on ts 3-dmensonal structure, every conformaton can vary n ts physcal, chemcal and bologcal property [Leach 2001]. As has been shown by Buggert [Buggert 2008], Jork [Jork 2005] and [Spuhl 2006], dfferent conformatons wll lead to dfferences n COSMO-RS actvty coeffcent predcton. Conformatons are 3-dmensonal arrangements of a molecule structure whch can be nterconverted by rotaton about sngle bonds [Mortmer 2001]. Intramolecular forces are responsble for conformaton formaton and due to small gaps n ther energy levels; they can easly blend nto each other. When lookng at a pure component, t becomes apparent, that the probablty of a conformaton occurrence solely depends on ts potental energy and the system temperature. Wthn a mxture, conformaton energy n soluton has to be consdered as well. Usng statstcal mechancs, probablty of occurrence can be descrbed by a Boltzmann dstrbuton [Leach 2001]. Hereafter, conformaton formaton shall be dscussed usng the example of an ethane molecule. By applyng the necessary force to move the hydrogen atoms passed each other, the ethane methyl groups can be turned aganst one another alongsde the C-C sngle bond. In case of ethane, the rotatonal barrer s so small that t can be spoken of free methyl group rotatablty [Vollrath 1990].

70 54 Bascs Fgure 2.10: Conformatons of ethane [Vollrath 1990] Fg dsplays the two extrema conformatons n respect to potental energy. Regardng the B projecton of Fg. 2.10, the ethane structure on the left sde shows a maxmum dstance between hydrogen atoms, leadng to a mnmum n potental energy. Consderng the rght hand sde of Fg on the contrary, hydrogen atoms appear to be lned up, causng an ncreased repulson of smlar charges and therefore leadng to a maxmum n potental energy. Between energy mnmum and maxmum les a rotaton of 60. Conformatonal search ams for fndng all possble molecule conformatons. The followng approach s at the bottom of all conformatonal searches: Based on a startng conformaton and by constant geometry varaton towards the pont potental energy mnmum, stable energy mnmum conformatons are tred to be detected [Leach 2001]. Fgure 2.11: Potental energy progresson of ethane conformatons [Chrsten 1972] Ethane wth ts sngle C-C bond, three stable energy mnmum conformatons exst. Ths number wll ncrease progressvely wth the number of avalable rotary bondngs. For nstance n-propane wth ts two rotary bondngs wll exhbt nne energy mnmum conformatons and

71 55 Bascs ths number wll ncrease to 243 wth the 5 rotary bondngs of n-hexane. From ths example t becomes obvous, that for ncreasngly large complex molecule structures, a sound and systematc search for stable mnma wll quckly lead to dsproportonal efforts. Therefore methods and algorthms have been developed, to tackle the challenge of complex molecule conformatonal search wth reasonable effort. A detaled revew on conformatonal search methods has been gven by Howard and Kollman [Howard 1988] and Leach [Leach 2001]. Wthn the COSMO-RS model, molecule conformatons are subject to an nternal weghtng procedure. Every nput conformaton wll therefore be provded wth a weghtng factor w. Calculaton of w wthn an arbtrary condensed phase s accomplshed as follows: w = j TFE TFE wc exp RT TFE j TFE wc j exp RT mn mn (2.111) The symmetry factor wc stands for the number of conformatons wth equal σ-profle. Usng the quantum chemcally calculated conformaton energy wthn an deal conductor E,COSMO, the screenng charge correcton E, as well as the chemcal potental of the conformaton wthn the condensed phase µ, the Total Free Energy (TFE) can be calculated [Klamt 1998]. TFE = E, + E + µ (2.112) COSMO 2.5 Models of statonary phases n RP-HPLC In lterature several models were conceved, whch am at reversed statonary phase descrpton n terms of composton and structure [Melander 1980]; [Martre 1983]; [Dll 1987]; [Carr 1993]; [Park 1993]; [Lochmüller 1979]. One example of reversed phase descrbng models s a model presented by Lochmüller and Wlder n 1979 [Lochmüller 1979]. The authors descrbe the reversed statonary phase as a lqud layer on the surface of the slca support. Ths lqud phase character and ts sorptve behavour s therefore formed by the bonded chans that tend to group to hydrophobc afflatons. Accordng to the authors, the exstence of a moble phase and a hydrophobc statonary phase wth lqud character wll lead to a partton lke elute dstrbuton mechansm (see Secton 2.2.7). Other authors found devatng forms of hypothetcal phase setups. The followng lnes are meant to gve a bref overvew on some of the exstng models.

72 56 Bascs Martre and Boehm [Martre 1983] modelled the reversed statonary phase n close accordance to the mentoned model of Lochmüller and Wlder as layer of lqud-crystallne hydrocarbon on top of the slca support. Here agan, elute dstrbuton s thought to be governed by a partton lke mechansm. Dll et al. [Dll 1987] brought forward two models that am for reversed statonary phase descrpton. One descrbes the bonded phase as amorphous-crystallne hydrocarbon layer that leads to elute parttonng, the other model conceves an elute adsorpton mechansm on top of laterally actng bonded alkyl chans. Horváth, Melander and Molnár [Melander 1980] understand elute retenton mechansm as dependent on the avalable contact area between elute and statonary phase molecules. In ther model, they descrbe the surface bonded chans as beng solvated and therefore stablzed wthn moble phase molecules. Kazakevch et al. [Kazakevch 2006] attempted to pcture the elute retenton process by a two stage model, where parts of the moble phase organc modfer wll adsorb as solvent layer on top of the reversed phase. The dstrbuton mechansm s then governed by () elute parttonng between the bulk moble phase and the adsorbed modfer layer as well as () elute adsorpton on the slca surface. Ths bref summary of exstng models llustrated the unsolved queston of detaled comprehenson. As has been mentoned before, reversed statonary phases are complex n character and the retenton mechansm wthn ths context s determned by many varable parameters.

73 57 Expermental 3 Expermental Ths secton wll gve an overvew on expermental methods as well as materal and equpment used to measure and model partton coeffcents. The frst secton s concerned wth materal and equpment that has been employed for retenton tme measurements at a chromatographc system. The second secton s meant to descrbe methods and programs used for modellng purposes together wth procedures for geometry optmsaton and conformatonal analyss. 3.1 HPLC materals, equpment and expermental methods Solvents (moble phase) Important moble phase selecton crtera were prevalence n analytcal and preparatve chromatographc applcatons, polarty and UV transparency. For test seres on nonpolar reversed statonary phases, aqueous mxtures of the polar organc solvents (modfer) methanol, tetrahydrofuran (THF) or acetontrle (ACN) were used to nvestgate polarty effects wthn a wde range of solvent polarty. For measurements on normal statonary phase, due to ts polar surface groups, used moble phase components had to be non-polar. In that case, varyng mxtures of n-heptane and cyclohexane were employed. HPLC-grade solvents were used for the preparaton of the moble phase. Before applcaton, water was dstlled, deonzed and fltrated usng a pore sze of 0.45 µm Solutes and tracer components Solute molecules (Tab. 3.1) were chosen by two major crtera: () the P O/W value, n order to valdate the developed models wthn a wde polarty range. The chosen substances cover a broad scope of the octanol-water partton coeffcent P O/W. () UV detectablty: To produce a suffcently good sgnal for peak analyss wth used UV/Vs-detectors, solute molecules had to exhbt a strong UV or vsble absorpton band. The system dead tme t 0 was evaluated wth njectons of the non-retaned organc marker uracl [Sander 1987]; [Wells 1981].

74 58 Expermental Table 3.1: Test solutes and assocated P OW values Nr CAS Solute P O/W p-bromotoluene ,2-Benzenedcarboxamde -1, Uracl -1, Pyrdnecarbontrle -0, Caffene -0, p-hydroxyacetanlde 0, Pyrdne 0, Asprn 0, Benzyl alcohol 1, Phenobarbtal 1, p-methylpyrdne 1, o-toludne 1, Hydroxyacetophenone 1, p-ntrophenol 1, Benzaldehyde 1, Acetophenone 1, Fluorophenol 1, Indazole 1, Phenylpropanol 1, P-Cresol 1, Ansole 2, Benzene 2, Propophenone 2, Fluorobenzene 2, p-ntrotoluene 2, Ethylbenzoate 2, Toluene 2, Chlorobenzene 2, Bromobenzene 2, p-xylene 3, m-xylene 3, Naphthalene 3, o-dchlorobenzene 3, Cumene 3, Propylbenzene 3, Acenaphthene 3, Butylbenzene 4, Pentamethylbenzene 4, Hexylbenzene 5, Naphthacene 5, n-decylbenzene 7, Statonary phases Concernng later applcablty of the developed predcton model, emphass was put on the descrpton of frequently used phases. After consultaton wth man manufacturer of analytcal and preparatve statonary phases, focus was set on C1, C4, C8 and C18 reversed phases (Tab. 3.2 to 3.4). To extend the predcton model towards reversed phases holdng actve groups, addtonal columns wth cyano and phenyl groups attached to the alkyl chans were acqured (Tab. 3.5). Columns packed wth polar and normal phase were purchased to perform NP-

75 59 Expermental HPLC measurements (Tab. 3.5). Tables 3.2 to 3.5 llustrate column propertes of all columns under nvestgaton. Table 3.2: Propertes of reversed statonary phase colums nvestgated wthn ths work Column name HyperslC18 Eclpse XDB-C18 Nucleosl C18 HD Abbrevaton HyperslC18 XC18 NuC18 Manufacturer Shandon HPLC, Runcorn, UK Hewlett-Packard Macherey-Nagel Dmensons L.d. [mm mm] 125 4, , Partcle sze [µm] 5 5 5,4 Pore sze [Å] Surface area [m²/g] Carbon loadng [%] 9,5 10,3 21 Surface coverage [µmol/m²] 3,5 3,6 Bulk densty [g/ml] 1 0,4 Bonded chemstry Trfunctonal Dmethyl-C18 Monomerc End cappng yes yes yes Table 3.3: Contnuaton of Tab. 3.2 Column name TSKgel ODS-2PW TSK-gel ODS-120T YMC-Pack C1 (TMS) Abbrevaton OD-2PW ODS-120T YMC C1 Manufacturer TosoHaas GmbH TosoHaas GmbH YMC Europe GmbH Dmensons L.d. [mm mm] 150 4, , ,6 Partcle sze [µm] Pore sze [Å] nm Surface area [m²/g] 330 Carbon loadng [%] 21 4 Surface coverage [µmol/m²] Bulk densty [g/ml] 1 0,4 Bonded chemstry Monomerc Monomerc End cappng no yes yes

76 60 Expermental Table 3.4: Contnauton of Tab. 3.3 Column name YMC-Pack Pro C8 YMC-Pack Pro C18 YMC-Pack Slca Abbrevaton YMC C8 YMC C18 YMC slca Manufacturer YMC Europe GmbH YMC Europe GmbH YMC Europe GmbH Dmensons L.d. [mm mm] 125 4, , ,6 Partcle sze [µm] Pore sze [Å] 12 nm 12 nm 12 nm Surface area [m²/g] Carbon loadng [%] Surface coverage [µmol/m²] Bulk densty [g/ml] Bonded chemstry End cappng yes yes Table 3.5: Contnuaton of Tab. 3.4 Column name Kromasl KR-60-5CN Phenomenex Prodgy phenyl-3 Restek Ultra C8 Abbrevaton Kromasl CN Prodgy phenyl Restek C8 Manufacturer Akzo-Nobel Phenomenex, Inc. Dmensons L.d. [mm mm] 150 4, , ,6 Partcle sze [µm] Pore sze [Å] Surface area [m²/g] 450 Carbon loadng [%] 10 Surface coverage [µmol/m²] Bulk densty [g/ml] Bonded chemstry Polymerc End cappng yes no

77 61 Expermental Chromatographc system setup A Htach LaChrom HPLC-System was used for all undertaken HPLC experments. For moble phase storage, the system contans two solvent reservors, whch are ndvdually connected to two membrane pumps (D7100). Before reachng the pump, the moble phase has to pass through a membrane degasser, where dssolved gas s removed by applyng vacuum. To hold back sold mpurtes, a flter s mplemented nto the ppng. The membrane pumps wth pulsaton suppresson wll apply the pressure necessary to overcome consecutve pressure loss by valves, ppng, column and detector. Subsequent to the pumps, the two moble phase streams are mxed usng a statc mxer and then conveyed towards a valve equpped wth a 100µl sample loop that allows for sample njecton. Injecton of solutes nto the valve s performed automatcally by an auto sampler (D7200), usng cuvettes sttng on an 8x10 matrx sample table. After transmgratng the valve, the moble phase stream flows nto the tempered column, stuated wthn a column oven (D7300). A dode array detector (DAD, L4500A) connected n seres measures the solute concentraton at the outlet of the column. All sensor sgnals (UV-detector, pressure gauges and thermometers) are transmtted to a personal computer va a data acquston nterface (D7000). Data recordng and analyss s done by Merck-Htach data processng software (D-7000 HPLC System Manager Verson 4.1) Expermental HPLC measurng methods Before every seres of measurements, the HPLC system was purged wth moble phase to flush out any gas bubbles that especally tend to accumulate wthn the pump valves. Gas bubbles that are accdentally conveyed nto a column can create vod spaces and therefore change column retenton behavour. The membrane pump nduced moble phase volume stream at system pressure s montored regularly to avod resultng errors n solute retenton tme. After each column mountng or dsmountng, the system was flushed wth moble phase for at least 30 mnutes and a flow rate of 1 ml/mn, to reach phase equlbrum between moble and statonary phase wthn the column and to drag out any remanng resdues. All test solutes were dssolved n moble phase accordng to ther solublty. Therefore, sample concentraton can vary between 0.2 to 2 mg/ml. To avod crystal formaton nsde a column due to local oversaturaton, not readly soluble substances were dluted from saturated soluton. All measurements were at least performed n trplcate to form an average and estmate the uncertanty. Detector sgnal valdty was ensured by perodcal detector wavelength calbraton.

78 62 Expermental k -factor determnaton from chromatographc measurements Accordng to Eq. 2.7, the k-factor k can be calculated va system dead tme t 0 and retenton tme t R, of the consdered elute. To obtan comparable k-factors wthn a seres of experments, the used dead tme marker as well as moble phase composton, system flow rate and temperature have to be kept constant. All expermental measurements wthn ths work were about k-factor estmaton. Therefore, nether elute concentraton nor ts peak area was of mportance. However, attenton was pad to the occurrence of a dstnct and symmetrcal elute peak Lterature research Lterature research bascally served a trple purpose: () the acquston of retenton data, () the achevement of a retenton model overvew and () and an overvew of statonary phase propertes. For acquston of addtonal expermental data of reversed and normal phase chromatographc systems, an extensve lterature research has been undertaken. The research has yelded a great abundance of data wth emphass on C8 and C18 systems. Most data s gven n form of k -factors, permttng the correlaton wth calculated partton coeffcents (Secton 2.2.5). A varety of publcatons wth accessble retenton data s gven n the bblography [Croes 2005]; [Glroy 2003]; [Jnno 1984]; [Sanag 1996]; [Tan 1996]; [Tsukahara 1993]. In the course of ths research, an overvew on dverse retenton models has been ganed and as a result the attempt has been undertaken, to gve a concse overvew on approaches used to model chromatography n general (see Secton 4.5 and Secton 2.3). Lterature nvestgaton also lead to a crude revew on technques that are capable to provde more drect evdence of bonded phase character n terms of lgand moton, conformaton and cooperatve assocatons (see Secton 2.1.8). 3.2 Computatonal modellng: programs and computatonal methods Molecule geometry generaton and conformatonal analyss All molecule geometres that have found applcaton wthn ths work were created usng the commercally avalable HyperChem program [HyperChem 2002]. Furthermore, all molecules were subject to a conformatonal analyss (Secton 3.2.5) also performed by HyperChem. The very frst step of the procedure, outlned n Secton 2.9 s the creaton of the three-dmensonal molecule structure usng a HyperChem mplemented drawng tool. The compled structure as well as all structures found n the course of conformatonal analyss wll be submtted to a

79 63 Expermental geometrcal optmzaton usng sem-emprcal quantum chemcal methods (here PM3 [Stewart 1990]) that leads to a local or global molecule energy mnmum calculated n vacuum. By applyng an advanced Monte Carlo multple mnmum (MCMM) Method [Chang 1989]; [Goodman 1991], a molecule start geometry wll be subject to systematcal varatons. By ths means, the molecule energy hyper area can be scanned wth gradually ncreasng resoluton. Hence energy level dfferences of addtonally found structures become ncreasngly smaller or n other words, the energy gradent between two conformatons steadly decreases, leadng to a growng number of conformatons. To lmt ths number, socalled acceptance crtera can be appled to each newly found conformaton. To perform a conformatonal search usng HyperChem, acceptance crtera must hence be defned and all rotaton angles wthn the molecule must be gven do determne the spectrum of structure varaton. The followng three acceptance crtera can be used to nfluence the number of found conformatons: 1) The Acceptance Energy Crteron Maxmum depcts the total energy dfference (n kcal/mol) to a currently exstng energy mnmum conformaton wthn whch a consdered conformaton s accepted. 2) The Energy dfference E (kcal/mol) between two conformatons. If the energy dfference between two conformatons falls short of ths value, the more recent conformaton wll be dscarded. 3) The root mean square error (RMS error). If the RMS error of two conformatons falls beneath ths adjustable parameter, the two conformatons wll be regarded as duplcate. The number of found conformaton s heavly dependng on molecule sze and complexty. It seems suggestve to lmt the number of conformatons to a physcally sensble level. Wth a lmted number of large and complex conformatons, an addtonal tme reducton regardng successve DFT calculatons can be acheved. Although t seems reasonable to take countermeasures aganst an ncreasng number of found conformatons by adjustng the acceptance crtera wth growng molecule sze and complexty, the author has chosen the approach of constant acceptance crtera values. Ths approach s founded by the search for a unversal conformaton selecton method (see Secton 3.2.5) that allows for the comparablty of all sets of conformatons. In contrast, the sem emprcal method PM3 was appled at all nvestgated molecule structures. PM3 also turned out to be tme consumng for bg and complex molecules. Conformatonal analyss of pseudo statonary phase molecules lke the one shown n Fg. 4.17, requred about 1 days of calculaton tme on a 3 GHZ dual core processor. Due to a

80 64 Expermental lmted number of such complex molecules, the tme nvested stll stayed wthn reasonable lmts Converson of HyperChem output data A complex molecule can result nto several hundred conformatons, wth each beng mapped n form of three dmensonal coordnates. Data of all conformatons of a sngle solute can be saved by Hperchem nto a sngle fle (solutename.hn). Space coordnates of all conformers are consecutvely wrtten wthn ths fle (Fg. 3.1). Fgure 3.1: Excerpt of.hn-data-fle. Atom coordnates of a caffene molecule To perform DFT calculatons, the three-dmensonal nformaton of every conformaton has to be wrtten nto a sngle.xyz-fle. Smultaneously, a coordnate transformaton from Bohr to Angstrom rad has to be performed. To date, ths procedure had been accomplshed by hand, consumng great amounts of tme. Usng the Vsual Basc (VB.NET) programmng software by Mcrosoft, the author developed a tool for automated fle converson. Bascally, ths tool accepts.hn fles and other HyperChem output fle formats as nput and converts them nto.xyz fles, holdng the three-dmensonal nformaton for successve DFT calculatons.

81 65 Expermental DFT geometry optmzaton Solutename.xyz fles serve as nput for DFT geometry optmzaton usng COSMO boundary condtons (Secton 2.4.3). All molecule structures used wthn ths work had to undergo ths calculaton step. DFT computatons are performed by the Turbomole software package [Ahlrchs 1989] verson 5.8, on an external Hgh Performance Computng (HPC) system. To obtan DFT calculatons on a hgh quantum mechancal level, the Trple Zeta Valence Polarzaton (TZVP) bass set [Echkorn 1997] s used whle the BP86 functonal [Perdow_1992] s appled to calculate the correcton term E XC (Eq. 2.93). Turbomole mplemented COSMO boundary condtons [Schafer 2000] allow for consderaton of a surroundng deal conductor. Despte the avalablty of hgh computng power, the sophstcated DFT calculatons for large and complex structures can lead to several days of computng tme. For ths reason, Turbomole verson 5.8 provdes the opton of a computaton parallelzaton. As a result, calculaton of large pseudo statonary phase molecules was feasble wthn 24 hours. DFT processng step output s the so-called COSMO fle (Solutename.cosmo). Hence, every nput.xyz fle wll result nto an output.cosmo fle nlcudng the optmzed structure nformaton and a molecule cosmo energy value. To enable char nternal access to.cosmo-fles, an ntranet database was establshed Calculaton of the chromatographc parttonng coeffcent usng COSMO-RS Parameters and data fles necessary for the parttonng coeffcent calculaton for nfnte solute dluton as defned n Eq. 2.73, are as follows: () compostons of solvent phases, () system temperature, () the used bass set, (v) the target value to be calculated and (v).cosmo fles of all components present n the system. The computaton s then performed wth the commercally avalable software COSMOtherm [COSMOtherm 2008]. The COSMOtherm program centrepece s an executve fle (cosmotherm.exe) that holds all necessary algorthms. Va command lne nterface, parameter values and fle locatons can be fed to the algorthm, whch wll eventually output the target value. A more easy to use program nterface s offered by COSMOthermX. Ths Java based COSMOtherm extenson can be run on a Wndows system and s meant to feed the cosmotherm.exe fle wth all necessary nformaton and subsequently chart the results after calculatons are completed. Wthn ths work, parttonng coeffcent K αβ s always calculated at nfnte solute dluton. Ths crcumstance s due to the exclusve consderaton of measurements from analytcal chromatographc systems. For such systems (wthn a lnear chromatographc regme), the amount of njected solute can assumed to be ndefntely dluted n the statonary and the moble phase (Secton 2.2.4). Bascally there are two ways to calculate K αβ usng

82 66 Expermental COSMOtherm. One s to compute the solute actvty coeffcents wthn each phase (γ α and γ β ) and subsequently buld the rato of both gammas. A shorter way s to drectly choose the log P-opton and set a value of 1 for the phase rato, so the log P value wll become a log K value (Eq. 2.50) Conformaton selecton for COSMO-RS calculatons For COSMO-RS calculaton nput, several conformatons can be used to depct a sngle molecule. Ths approach of conformaton consderaton leads to a remarkable ncrease n smulaton effort and can cause the algorthm to crash f the lmted number of processable surface segments s exceeded. Nevertheless, compared to DFT calculatons, tmeconsumpton of the effcent COSMO-RS algorthm s stll qute manageable and wll not exceed several mnutes. As shown n Secton 2.4.6, the consderaton of several conformatons s meant to better descrbe the real physco-chemcal characterstcs of a molecule. A COSMO-RS mplemented weghtng factor w (Eq ) wll nternally determne the contrbuton strength of each conformaton. Thus the queston arses, whch conformaton complaton has to be chosen to best represent the molecule. In practce several selecton approaches exst. The smplest approach s to choose the sngle lowest energy conformaton. Some authors wll not choose more than 10 conformatons, amng to fnd a method of conformatonal search that wll lmt found conformatons to ths number. A thrd method s to pck an energy weghted average from the whole conformaton spectrum. The nternal weghng factor wll than lead to an overrepresentaton of the chosen low energy conformatons. The author has compared methods and subsequently appled the equal E COSMO energy dfference method to all calculatons wthn ths work Selecton of conformatons wth equal dfference n E COSMO As has been explaned n the secton above, no unform or well establshed method for conformaton selecton exsts to date. Due to ths non-exstence of a system dependent default selecton rule, the author has defned a consstent approach that apples to any system consdered wthn ths work. The procedure s structured nto the followng steps: 1. Conducton of the HyperChem conformaton search. 2. DFT analyss of the found compound conformatons. 3. Creaton of a COSMO energy (E COSMO ) lst. 4. Selecton of 10 conformatons. The conformatons are selected wth constant COSMO energy nterval whle coverng the total found energy spectra.

83 67 Expermental To create a COSMO energy lst, the range n E COSMO s gathered from the whole conformaton spectra. E COSMO values can be found wthn the.cosmo fles. These fles are the DFT output fle format. Snce a manual E COSMO value extracton can be very tme consumng, dependng on the number of exstng conformatons, a Vsual Basc tool (Fg. 3.2) was programmed by the author to automate ths procedure. Fgure 3.2: Work screen of energy lst creator tool

84 68 Results & Dscusson 4 Results & Dscusson Elute partton coeffcent predcton wthn a lqud chromatographc system depcts the goal of ths work. The queston wll be addressed, f elute dstrbuton between a moble and statonary phase can be predcted by the COSMO-RS model. In ths context, the feasblty of statonary phase smulaton usng a pseudo-lqud statonary phase molecule wll be dscussed and dfferent parameters wll be nvestgated regardng ther nfluence on predcton accuracy. Bascally all nvestgatons presented below shall lead to an approach, capable to model chromatographc elute parttonng n order to smplfy the choce of a separaton task optmzed phase system. The frst secton deals wth methods of conformaton selecton. For ths, the well-known octanol-water system wll be consulted and expermental data wll be compared to calculaton results to evaluate the nfluence of solute and solvent conformaton consderaton. The second secton addresses the modellng of reversed and normal phase statonary phases, usng pseudo-lqud molecules. Evoluton n phase modellng wll be presented and calculaton result are compared and valdated wth expermental data. As a result of an extensve lterature research, the last secton wll gve an overvew on nterrelatons of emprcal retenton predcton models. 4.1 Conformaton selecton of solute and solvent molecules: Influence on calculaton results / Selecton rules Effect of conformaton selecton on γ The goal of a conformatonal analyss s the dentfcaton of those conformatons, whch determne the molecule nteracton characterstcs wthn a gven system. The mpact of conformaton selecton on γ s to be presented below. Before a set of conformaton can be chosen, a number of conformatons have to be found usng conformatonal analyss (Secton 3.2.6). Such a conformatonal analyss was conducted for a molecule structure named decyldmethlyslanol. Among others, ths molecule was used to smulate the statonary phase and shall serve as example for the explanatons below. 357 conformatons were found for decyldmethlyslanol. All conformatons were subject to DFT geometry optmzaton usng COSMO boundary condtons (Secton 2.4.3). For ths set of conformatons, the calculated COSMO energy E COSMO vares wthn a range of -839,169 and -839,193 kj/mol. In the followng setup, decyldmethlyslanol served as solvent molecule whle phenol was used as

85 69 Results & Dscusson solute. The COSMO-RS calculated actvty coeffcents of phenol γ Phenol at 298 K and nfnte dluton wthn the solvent are pctured n Fg Here, the actvty coeffcent depends on the choce n solvent conformaton, whle for all computatons, the solute structure has been kept constant. Resultng γ Phenol were plotted aganst the free energy of decyldmethlyslanol conformatons n soluton. 0,5 0,4 γ nf phenol 0,3 0,2 0,1 0,0-807, , , , ,606 E COSMO of solvent conformaton 10³ [kj/mol] Fgure 4.1: Plot of γ Phenol (solute) over E COSMO of decyldmethylslanol conformatons (solvent).the horzontal lne depcts γ Phenol wth a selecton of decyldmethylslanol conformatons from Tab. 4.1 used for calculaton. In dagram 4.1, γ Phenol values vary wthn a range of 0,028 and 0,461. The horzontal lne at γ Phenol = 0,189 depcts the phenol actvty coeffcent value calculated wth a solvent composed of a decyldmethylslanol conformatons selecton (Tab. 4.1). The nternal weghng of conformaton contrbutons has been conducted by Eq Tab. 4.1 represents 10 selected conformatons wth equal dfferences n E COSMO. Ths selecton has been chosen to cover the total found energy range. For explanaton of the chosen equal dfferences conformaton selecton approach see secton

86 70 Results & Dscusson Table 4.1: Decyldmethlyslanol conformaton selecton. Total COSMO energy range s presented by ten conformatons wth constant energy ntervals of about -2,3 J/mol. Conformaton number E COSMO [-kj/mol] 839, , , , , , , , , ,1718 Fgure 4.2: σ-profles and assocated structures of decyldmethlyslanol conformatons The structures llustrated above the correspondng σ-profles (Fg. 4.2) depct conformatons of the decyldmethylslanol molecule (Tab. 4.1). Wth an ncreasng E COSMO value, the conformaton structures wll change from a more lnear to a bent (non-lnear) shape, whle the probablty of occurrence p(σ) at zero surface segment charge (σ = 0) wll decrease

87 71 Results & Dscusson tendentally. Ths leads to the concluson that n case of alkyl chans, ntramolecular repulsve forces become smaller wth a more lnear shape. As the probablty of occurrence of a certan structure s expressed by the Boltzmann dstrbuton wth respect to ts potental energy, the relatve porton of lnear conformatons wll preval f regardng a system at vacuum condtons. If a dssolved molecule s regarded, specfc conformatons wll be stablzed and others destablzed, accordng to ther nteracton wth surroundng solvent molecules [Nkk 2001]. In dssolved state, addtonal to the Boltzmann dependng potental energy, solvent nteractons and temperature dependence have to be consdered. The mpact of solute-solvent nteractons on solute actvty coeffcents at nfnte dluton shall be llustrated at two dfferent solvent systems along wth four dssolved solute molecules, as there are chlorobutane, ntropentane, hexane and heptanol. A polar and a non-polar solvent system were chosen, consstng of an equmolar methanol/water mxture and pure decyldmethylslanol, respectvely. Tab. 4.2 shows the actvty coeffcent at nfnte dluton γ for dfferent solute conformatons and γ of a selecton of 10 conformatons (average γ ), respectvely. Tab. 4.2 values were calculated usng a constant solvent molecule conformaton ensemble n each case, whle varyng the selected solute conformatons. Table 4.2: Solute actvty coeffcents of maxmum/mnmum conformatons (γ ) and conformaton selecton (γ ) at nfnte dluton Solvent: Pure decyldmethylslanol (conformaton ensemble accordng to Tab. 4.1) Solute Maxmum γ Mnmum γ Average γ 1-Chlorobutane 1,39 1,37 1,39 1-Ntropentane 2,60 2,35 2,56 1-Hexane 1,42 1,38 1,40 1-Heptanol 1,11 0,71 1,10 Solute Solvent: Methanol/water mxture of 1:1 molar composton Maxmum γ Mnmum γ Average γ 1-Chlorobutane 38,96 33,19 37,48 1-Ntropentane 43,76 28,27 39,34 1-Hexane 142,43 106,29 140,36 1-Heptanol 10,69 7,21 9,54

88 72 Results & Dscusson Table 4.3: Conformaton structures for γ mn and γ max values, respectvely. Infnte solute dluton n equmolar methanol-water mxture Structure of max γ -value n solvent: methanol/water mn γ -value n solvent: decyldmethylslanol Structure of mn γ -value n solvent: methanol/water max γ -val. n solvent: decyldmethylslanol Conformaton 1 of 3 Conformaton 3 of 3 1-Chlorobutane 1-Ntropentane Conformaton 1 of 7 Conformaton 7 of 7 1-Hexane Conformaton 1 of 5 Conformaton 5 of 5 1-Heptanol Conformaton 1 of 24 Conformaton 22 of 24 As explaned above, lnear alkyl chan structures wll have the lowest potental energy and therefore the hghest probablty to exst n vacuum. Applcaton of ths understandng on conformaton selecton n dssolved state wll lead to extrema n calculated γ. For non-polar solutes dssolved wthn non-polar solvents, the choce of lnear conformaton structures wll for the most part lead to a mnmum n solute actvty coeffcent, whle wthn polar solvents a maxmum n γ wll result (see Tab. 4.2). Ths phenomenon can partly be explaned by an entropy effect of the solvent molecules. When lookng at Tab. 4.3 t becomes obvous that lnear structures dssolve better (smaller γ) wthn unpolar solvents than bent structures and vce versa. Gven that bent structures offer a smaller solvent accessble surface than lnear ones, the lne of argument also holds true f usng area nstead of polarty. A smaller accessble area results nto a smaller solute-solvent nterface. Therefore, less solvent molecules suffer a loss n ther degrees of freedom because of boundary effects. So f the nterface area s decreased, system entropy wll ncrease. Accordng to Gbbs Helmholtz equaton, entropy ncrease leads to a decrease n system free energy. Wthn the COSMO-RS

89 73 Results & Dscusson model, these dfferences between conformatons are expressed wth varatons of σ-profles and resultng msft energes between solute and solvent screenng charges (Secton 2.4.4). Fgure 4.3: Sgma profles of dfferent solutes The structural dfferences of conformatons shown n Tab. 4.3, are represented n Fg. 4.3 va σ-profles. Structural changes wll lead to changes n surface segment charges σ, depcted as probablty dstrbuton p(σ) (Secton 2.4.4). An alternatve way to pont out the mportance of conformaton selecton s shown by means of a homologous row of alcohols. If consderng the γ ranges resultng from the conformaton spectra, t becomes obvous, that the possblty exsts, to mx up the homologous row by choosng the wrong conformatons for calculaton.

90 74 Results & Dscusson Fgure 4.4: γ ranges of dfferent alcohols resultng from ther conformaton spectra. The solvent phase conssts of an equmolar water/methanol mxture. Although Fg. 4.4 s able to reflect the correct trend n solublty, t s stll a challenge to choose a meanngful solute and solvent conformaton ensemble wthn a real system. And snce the calculated actvty coeffcent strongly depends on the chosen conformaton ensemble, computatonal descrpton of a real system needs rules for conformaton selecton that are valdated by expermental data. A rule for conformatonal selecton as descrbed n Secton wll be developed and valdated n the followng sectons Effect of solute and solvent conformaton selecton on K OW The octanol-water partton coeffcent (K OW ) s used to descrbe the equlbrum dstrbuton of a substance between an octanol rch and a water rch phase. As explaned n Secton 2.2.5, the partton coeffcent K αβ of a component between two phases can be expressed by the rato of the compound actvty coeffcents at nfnte dluton wthn both phases. Bass of K αβ predcton s a sutable conformaton selecton concernng all system components as well as the correct choce of coexstng phase compostons. To calculate the octanol-water partton coeffcent and subsequently enqure comparsons wth expermental data, the effect of two possble nfluence parameters wthn the solvents has to be consdered. () an nput parameter that can vary s the phase composton of the octanol rch and the aqueous phase, respectvely. () secondly, the sets of selected octanol conformatons can dffer. Both varatons lead to changes n COSMO-RS calculaton results, as can be seen n Tables 4.6 and 4.7. The nfluence of the selected conformaton set wll be dscussed n the followng secton. As to model valdaton, the octanol-water system has been chosen for dfferent reasons: On the one hand, the aforesad system has great resemblance to reversed phase chromatographc

91 75 Results & Dscusson systems under nvestgaton what wll be of nterest n the secton to follow; on the other hand, the octanol-water system s well examned wth varous expermental data exstng n lterature. If not hghlghted dfferently, the octanol-water phase composton for the COSMO-RS calculatons wthn ths secton was conducted from [Dallas 1992] as follows: Table 4.4: Composton of the two coexstng phases n the n-octanol-water system at 25 C Octanol rch phase Aqueous phase x n-octanol x water x n-octanol x water [Dallas 1992] 0,711 0,289 0, , To nvestgate the dependency of the octanol-water partton coeffcent on dfferent n-octanol conformatons, K OW was calculated for several solute molecules (Tab. 4.6). For each solute out of Tab. 4.6 and Tab. 4.7, the molecule structure was held constant, whle COSMO-RS calculatons were performed wth 6 dfferent octanol conformatons as well as one octanol conformaton selecton (Tab. 4.5). As water only has one conformatonal structure t s used unchanged wthn all calculatons. Table 4.5: Octanol conformaton selecton. Total COSMO energy range s presented by ten conformatons wth constant energy ntervals of about 0,0016 kj/mol Conformaton number of n- octanol E COSMO [-kj/mol] 391, , , , , , , , , ,0569

92 76 Results & Dscusson Table 4.6: Expermental and COSMO-RS calculated K OW at T = 25 C; Varable nput parameter for calculaton was the n-octanol conformaton; Expermental data taken from [Sangster 1997] and [Harnsch 1983]; The COSMO RS exp exp relatve error δ was calculated as follows: δ (log K) = 100( log K log K log K ) Octanol Conformaton Solute Exp. log K OW log K OW δ / % log K OW δ / % log K OW δ / % log K OW δ / % 1,1,1,2-Tetrafluoroethane 1,88 1,97 4,90 1,96 4,19 2,03 8,06 2,18 16,08 1,1,1-Trchloroethane 3,27 3,35 2,33 3,35 2,42 3,37 2,95 3,42 4,56 1,1-Dfluoroethane 1,52 1,81 18,78 1,82 19,44 1,83 20,63 1,91 25,74 1,2-Dchloroethane 2,25 2,47 9,61 2,45 8,90 2,51 11,35 2,59 14,95 Buthylbenzene 5,03 3,76 25,18 3,88 22,84 3,77 24,96 3,80 24,38 Decylbenzene 4,19 5,18 23,67 5,22 24,50 5,20 24,06 5,27 25,76 Dchlorodfluoromethane 7,37 8,44 14,45 8,48 15,10 8,45 14,68 8,54 15,89 Dchloromethane 2,88 3,05 5,86 3,07 6,68 3,06 6,20 3,10 7,73 Ethane 2,05 2,14 4,47 2,10 2,43 2,19 7,04 2,30 12,27 Ethene 2,49 2,59 3,91 2,60 4,42 2,59 3,97 2,61 4,72 Hexylbenzene 1,97 2,08 5,81 2,10 6,56 2,09 6,23 2,12 7,86 Methane 5,25 5,71 8,79 5,75 9,50 5,73 9,11 5,80 10,52 p-ntrophenole 1,89 1,97 4,27 1,98 4,80 1,97 4,36 1,99 5,18 Trchloroethene 1,88 2,15 14,39 2,04 8,27 2,28 21,16 2,52 33,84 Trchlorofluoromethane 3,14 3,53 12,52 3,52 12,25 3,56 13,44 3,63 15,75 Trchloromethane 3,24 3,44 6,25 3,46 6,80 3,45 6,51 3,49 7,66 Trfluoromethan 2,76 2,99 8,39 2,94 6,55 3,05 10,35 3,16 14,36 Average relatve error 10,29 9,73 11,71 15,13 Table 4.7: Contnuaton of Tab. 4.6 Octanol Conformaton Selecton (Tab. 4.5) Solute Exp. log K OW log K OW δ / % log K OW δ / % log K OW δ / % log K OW δ / % 1,1,1,2-Tetrafluoroethane 1,88 1,97 4,57 1,92 2,30 1,97 4,59 1,94 3,30 1,1,1-Trchloroethane 3,27 3,33 1,79 3,37 2,93 3,33 1,77 3,31 1,34 1,1-Dfluoroethane 1,52 1,82 19,58 1,83 20,50 1,82 19,64 1,78 17,34 1,2-Dchloroethane 2,25 2,44 8,34 2,45 8,99 2,44 8,26 2,44 8,28 Buthylbenzene 5,03 3,90 22,43 4,19 16,76 3,90 22,49 3,78 24,93 Decylbenzene 4,19 5,19 23,96 5,26 25,63 5,20 23,99 5,14 22,72 Dchlorodfluoromethane 7,37 8,44 14,46 8,55 16,02 8,43 14,44 8,38 13,65 Dchloromethane 2,88 3,06 6,27 3,09 7,38 3,06 6,37 3,02 4,89 Ethane 2,05 2,09 1,91 2,06 0,36 2,08 1,68 2,11 2,85 Ethene 2,49 2,59 3,88 2,61 4,91 2,59 3,88 2,56 2,89 Hexylbenzene 1,97 2,09 6,27 2,12 7,48 2,10 6,36 2,06 4,69 Methane 5,25 5,72 8,99 5,80 10,46 5,72 9,00 5,67 7,97 p-ntrophenole 1,89 1,97 4,32 1,99 5,25 1,97 4,33 1,95 3,12 Trchloroethene 1,88 2,05 8,80 1,80 4,12 2,07 9,92 2,11 12,37 Trchlorofluoromethane 3,14 3,50 11,62 3,52 11,96 3,50 11,58 3,50 11,36 Trchloromethane 3,24 3,44 6,22 3,48 7,44 3,44 6,26 3,41 5,30 Trfluoromethan 2,76 2,92 5,83 2,86 3,80 2,92 5,70 2,95 6,91 Average relatve error 9,40 8,96 9,46 9,08

93 77 Results & Dscusson For the solute set under nvestgaton, the qualty of predcton shows a dependency on the used octanol conformaton, as can be seen from the varatons n relatve errors of Tables 4.6 and 4.7. Accordng to Tab. 4.5, the octanol conformatons are arranged by ther E COSMO. Even so, no dstnct correlaton between the mean absolute error and the E COSMO can be observed from Tables 4.6 and 4.7. Nonetheless, startng from a certan degree of molecule entanglement (conformaton 33), the error values depcted n Fg. 4.5 wll show no more sgnfcant varaton. When comparng calculaton results, t was found that for the solute set under nvestgaton, the mean absolute error of predcton for the weghted conformaton selecton s arranged among the three lowest values (along wth conformaton 39 and 46) and on a par wth the expermental average standard devaton [Harnsch 1983]. Furthermore, the same set of selected conformatons that has not undergone a COSMO-RS nternal weghtng procedure (Secton 2.4.6) wll exhbt a somewhat greater relatve (0,32 unts greater) and absolute (0,01 unts greater) error, when compared to the weghted set of conformatons. Fgure 4.5: Mean absolute errors of predcted solute K OW values at T = 25 C; Varable nput parameter (x-achss) s the octanol conformaton; the dotted lne shows the expermental average standard devaton conducted from [Harnsch k 1983]; 1 exp COSMO RS = log K log K k = 1

94 78 Results & Dscusson Another dependency can be found n the water content wthn the octanol-rch phase. In lterature, the octanol-water system has often been subject to expermental examnaton [Buggert 2008] and therefore, dfferent values can be found for the water content of the octanol-rch phase. Values can reach from x water = 0,207 [Sörensen 1979] to x water = 0,289 [Dallas 1992]. In Fg. 4.6, the two mentoned values can approxmately be depcted by columns 3 and 4. For the solute set and the octanol selecton used for log K calculaton, the average relatve error shows clear dependency on the x water value. Fgure 4.6: Average relatve errors of predcted solute K OW values at T = 25 C; Average relatve error wth k depctng the number of solute molecules ; k 1 δ = log K k = 1 COSMO RS log K exp log K exp 100%

95 79 Results & Dscusson Fgure 4.7: Comparson of COSMO-RS calculated and expermental K OW values (see Tables 4.6 and 4.7). Impact of the octanol conformaton used to predcted solute K OW values at T = 25 C. Lookng at Fg. 4.7, t s possble to dentfy a sngle conformaton that allows K OW predcton n the same mean absolute error range as f a conformaton selecton s beng used. But on the other hand, t s also possble to choose a sngle conformaton that wll lead to a much hgher mean absolute error (Fg. 4.6). Therefore, expermental data has to be consulted before the rght sngle conformaton can be chosen. As COSMO-RS calculatons am for predcton n order to avod expermental effort, t becomes obvous that choosng a conformaton selecton s the superor and more relable approach Dervaton of a conformaton selecton rule for solute and solvent molecules As has been shown n the secton above, conformaton selecton has consderable effect on COSMO-RS calculaton results regardng phase behavour predcton. Investgatons by Buggert et al. [Buggert 2009] and more recently by Mokrushna et al. [Mokrushna 2012]

96 80 Results & Dscusson have shown that conformatons generated by molecular dynamc smulaton methods wll lead to yet unsurpassed K predcton qualty. Stll to date no well-establshed method exsts to carry out such a selecton n advance. Therefore, the way that has been chosen by the author reles on a prncpal selecton procedure, whch can be appled to any solute and solvent under consderaton. The more complex ssue of smulatng the slca bound alkyl chans of the statonary phase and consequental approaches of molecule generaton and conformaton selecton wll be addressed n Secton The prncpal reason for a unfed selecton procedure s to guarantee comparablty of all COSMO-RS calculatons wthn a study or even between dfferent studes. In the followng, ths procedure shall be explaned and valdated by reference to expermental data Determnng the moment of conformaton selecton wth regard to the process step Before performng a conformaton selecton, one has to be aware of the fact that a selecton of conformatons can be done at three dfferent stages wthn the calculaton process: 1. Selecton by settng hard acceptance crtera: The frst opportunty to select a set of conformers arses, when adequate acceptance crtera for the conformatonal analyss are beng set. As explaned n Secton 3.2.1, HyperChem acceptance crtera can be set to reduce the number of found conformatons. For molecule structures as they are used to smulate the statonary phase, HyperChem conformatonal analyss can consume up to several days on a Pentum 4 dual core processor and result n many hundreds of conformatonal structures. But f applyng a unversal analyss approach, n whch comparablty of all conformaton sets s acheved by constant parametersaton, ncreased calculaton tme has to be accepted. Wthn the bounds of ths work, wth the excepton of some more complex statonary phase structures, most molecules were of manageable complexty and therefore the calculaton effort stayed acceptable. Varaton of acceptance crtera s an effectve way to lmt the number of found conformatons. Many authors have fallen back on ths approach to create ther ntal sets of conformatons. 2. Selecton by pckng before DFT calculaton: Another opportunty to choose a set of conformatons s the conformaton spectra tself, after beng generated by HyperChem. Wthn such a conformatonal analyss, potental energy s assgned to every created molecule structure. Ths energy spectra can be used to select a fxed number of conformatons by applyng a constant selecton procedure (Tab. 4.1). Ths approach can be used to complement

97 81 Results & Dscusson the method of constant acceptance crtera. Therefore, conformaton number lmtaton to a fnal set s not done by changes n acceptance crtera but by a constant selecton procedure after HyperChem conformatonal analyss. 3. Selecton by pckng after DFT calculaton: The thrd way to select a fnal set of conformatons s resembled by the approach depcted n the prevous secton (Secton 3.2.6, pont 2). By passng the complete HyperChem output conformaton set through the DFT geometry optmzaton step, a COSMO energy (E COSMO ) wll be computed for every structure. The fnal set of conformatons s then selected from the E COSMO -spectra accordng to the procedure depcted n Secton A dstncton between the conformatons created by HyperChem and the DFT processed conformatons can be found n the potental energy assocated wth the partcular conformatons. Fg. 4.8 shows the E COSMO -spectra output lst when usng the VB tool (Fg. 3.2), whle Fg. 4.9 depcts the choce of conformatons wth equal energy nterval accordng to Secton 3.2.6, pont 4. Fgure 4.8: E COSMO lst of 1-ocanal conformatons 4. All n: A fourth way to take all found conformatons (wthout usng hard acceptance crtera) s most reasonable but stll lmted n ts practcablty by a maxmum number of conformatons to be used wthn COSMO-RS calculatons. To date ths procedure s practcable for systems wth small molecules and a manageable total number of conformatons.

98 82 Results & Dscusson Fgure 4.9: Plot of E COSMO lst from Fg.4.8; 10 conformatons hghlghted for selecton. Wth Fgure 4.10, dfferences n HyperChem and DFT calculaton results can be llustrated. As a result of HyperChem conformaton analyss, conformatons are named and numbered accordng to ther potental energy calculated by HyperChem. Whle name and numberng s not changed, the energy of each conformaton s recalculated under DFT boundary condtons. From Fg. 4.9 t can be seen that the lst-poston due to E COSMO does not match wth HyperChem numberng. Otherwse the x-axs dfference between two consecutve red ponts would have to be constant. Ths demonstrates that results of potental energy calculaton under vacuum condtons dffer from DFT calculaton results, whle the overall tendency remans the same. A comparson of selecton methods 2 and 3 can be depcted by usng σ-profles (see Fg 4.10). Profle 1 shows an average σ-profle of the 10 chosen conformatons of Fg. 4.9 whle profle 2 shows an average σ-profle generated by 10 conformatons chosen wth constant energy ntervals from the orgnal HyperChem conformatonal analyss output lst.

99 83 Results & Dscusson Fgure 4.10: Sgma profle calculated from 1-octanal conformatons. As can be seen from the fgure above, both σ-profles overlap over a broad range of ther curve shape. Ths leads to very smlar results n any further statstcal COSMO-RS calculaton steps. Therefore, regardng the example of 1-octanal, method 3 wll not be able to sgnfcantly outperform method 2. Nevertheless, the author has chosen the latter approach and therefore selects the fnal set after all conformatons have passed the DFT geometry optmzaton. Choosng conformatons from an E COSMO lst s more meanngful due to the fact that the systems under nvestgaton are present n lqud state and therefore should be better represented by a DFT calculaton approach. Another great advantage les n the fact that the whole conformaton spectrum exsts n the DFT processed state. Therefore, changng the fxed number of selected conformatons wll not requre addtonal DFT calculaton. Furthermore a complete conformaton set s avalable to other researches va an nternal data base. 4.2 Development of reversed statonary phase modellng As has been shown n Secton 2.3, several models exst n lterature that am to descrbe the complex behavour statonary phases n HPLC. The hypothess and tools behnd the approach that wll be developed, appled and valdated n the followng secton can be found n detal n Sectons and 3. Bascally, results gven below shall gve an answer to the followng four questons: () s t possble to predct statonary phase behavour wth COSMO-RS

100 84 Results & Dscusson calculatons combned wth the use of pseudo-lqud molecules? () f so, s there a standard procedure to create such pseudo molecules that wll gve an average good system predcton? () where are lmtatons to ths approach regardng reversed statonary phases and (v) s t magnable to chemcally defne new statonary phases from mere calculatons based a-pror phase optmzaton? Development of pseudo-lqud molecules COSMO-RS calculaton of the partton coeffcent usng a smulated pseudo-lqud phase was motvated from the percepton, that over broad ranges, lqud chromatographc partton coeffcents can be correlated to lqud-lqud partton coeffcents. Tsukahara et al. [Tsukahara 1993] compared lqud-lqud solute parttonng between non-polar dodecane and a methanol-water mxture wth solute retenton data of a RP-HPLC system usng an ODS column (TSK-gel ODS-120T; Tab. 3.3) wth ts slca surface bound unpolar C18 alkyl chans as quas non-polar phase. As can be seen from Fg. 4.11, Tsukahara et al. has used the expermentally estmated partton coeffcent (log K exp,tsukahara ) of the dodecane methanol/water phase system to the k -factor from chromatographc measurements (log k exp ) for comparatve purpose. As to the COSMO-RS calculatons, a conformer selecton of the dodecane molecule was used to smulate the non-polar phase and to predct the log K under equal system condtons. The polar phase was smulated wth methanol and water molecules n analogy to expermental settngs. The dodecane COSMO-RS predctons are n good agreement wth results from Tsukahara et al. (Fg. 4.11).

101 85 Results & Dscusson Fgure 4.11: Correlaton between expermental log K, expermental log k and COSMO-RS predcted log K values of the dodecane-methanol/water and dodecanol - methanol/water systems at T = 25 C. Expermental values were taken from [Tsukahara 1993]. Values see Tab Wth ncreasng solute polarty and therefore decreasng log K values, system smulaton usng dodecane shows ncreased devaton from the lnear plot (Fg. 4.11). Expermental as well as COSMO-RS predcted log K values therefore show decreased compatblty wth the assumed lnear relatonshp between log K and log k, as t s expressed n Eq Accordng to Tsukahara et al., an upward devaton from the lne mples that solutes wth low hydrophobcty (.e. the more polar ones) have a hgher solublty n the statonary phase than n dodecane. Ths agan leads to the concluson that a statonary ODS phase has more polar characterstcs than lqud dodecane.

102 86 Results & Dscusson Table 4.8: Expermental and COSMO-RS calculated log K values for a system consstng of dodecane and a methanol-water mxture at T = 25 C; Water content x water of the polar methanol-water phase s 0,2. Varable calculaton nput parameter s the solvent molecule; Expermental data taken from [Tsukahara 1993]; The relatve error δ [%] s calculated as COSMO RS exp exp follows: δ [%] = 100% ( log K log K log K ) Solvent Dodecane Dodecanol Decyldmethylslanol Solute exp. logk logk δ / % logk OW δ / % logk OW δ / % 2-Phenylethanol -0,78-0,67 14,57 0,08 110,17 0,08 110,29 Acetophenone 0,18 0,20 6,91 0,41 123,85 0,43 133,23 Benzene 1,05 0,83 20,30 0,79 24,60 0,84 19,35 Benzophenone 0,62 0,49 20,49 0,62 0,03 0,66 6,89 Benzylalcohol -0,99-0,62 37,06 0,04 104,36 0,04 104,16 Chlorobenzene 1,12 0,89 20,82 0,85 24,41 0,91 19,18 Decane 2,59 2,45 5,31 2,13 17,82 2,20 14,93 Ethylbenzene 1,39 1,20 13,38 1,09 21,37 1,15 16,92 Ethylphenylether 1,07 0,95 11,27 0,91 14,77 0,97 9,32 Hexane 1,92 1,75 8,77 1,55 19,53 1,60 16,42 Methyl benzoate 0,56 0,50 9,57 0,61 9,79 0,65 17,27 Naphthalene 1,25 0,95 24,23 0,88 29,50 0,95 24,31 Ntrobenzene 0,36 0,42 15,86 0,51 42,39 0,57 57,86 Octane 2,23 2,10 6,02 1,83 17,88 1,90 14,87 p-dchlorobenzene 1,27 0,92 27,81 0,90 29,05 0,96 24,39 p-xylene 1,45 1,76 21,66 1,54 6,60 1,60 10,71 Propyl benzoate 0,94 0,90 4,41 0,94 0,38 0,98 5,14 Toluene 1,25 1,64 31,15 1,45 15,59 1,51 20,11 Average relatve error 16,64 34,00 34,74 Tab. 4.8 show dfferences n the average relatve error between log K exp and log K COSMO-RS due to dfferent solvents beng used for COSMO-RS calculaton. As can be seen from Tab. 4.8 and Fg. 4.12, best correlaton between predcted and expermental log K values s reached when dodecane s used as solvent for calculaton nput. If solvent molecules that exhbt a more polar characterstca are use for predcton, a poorer correlaton wll result. Ths leads to the concluson that COSMO-RS s senstve to solvent polarty and expermental values generated wth a non-polar solvent as dodecane wll be best predcted, f an analogous molecule s beng used for calculaton.

103 87 Results & Dscusson COSMO-RS Fgure 4.12: Average relatve error of predcted log K values for a LLE system consstng of dodecane and a methanol-water mxture at T = 25 C; Water content x water of the polar methanol-water phase s 0,2. Varable calculaton nput parameter s the solvent molecule. Expermental data was taken from [Tsukahara 1993]. The average relatve error wth k (k=18) depctng the number of solute k molecules s calculated as follows: 1 COSMO RS exp exp δ ( log K log K log K ) 100% = k = 1 Fgure 4.13: Correlaton coeffcent between log K COSMO-RS and Log k exp values for a RP-HPLC system consstng of a statonary ODS phase and a methanol-water moble phase at T = 25 C; Water content x water of the polar methanol-water phase s 0,2; Expermental log k data was taken from [Tsukahara 1993]

104 88 Results & Dscusson From Fg t becomes obvous that predcton of Log k exp values gve hgher correlaton coeffcents, f solvents wth polar character are used for smulaton of elute-statonary phase nteractons wthn the RP-HPLC system. Ths fndng correlates wth the concluson of Tsukahara et al. that a statonary ODS phase exhbts more polar characterstcs than dodecane. Ths result s further underpnned by the assumpton that polar, not suffcently shelded slanol groups bound to the surface of the slca support wll nteract wth elutes [Gmehlng 1998]. Many authors also assume that organc modfer such as methanol or acetontrle adsorb onto the statonary phase and therefore add polar characterstca to the statonary phase. In summary, analyss of Fgures 4.11 to 4.13 leads to three dfferent nferences: () COSMO- RS s able to predct log K between lqud phases n good accordance to the expermental values over the complete solute polarty range. () Dodecane as model-molecule s not ft to smulate all types of nteractons wthn a chromatographc system. () Usng solvent molecules wth polar characterstcs, COSMO-RS s able to better model a reversed statonary phase than the lqud-lqud approach of Tsukahara et al. From concluson () comprehenson was ganed that COSMO-RS calculatons along wth approprate solvent (pseudo-lqud) molecules are fundamentally capable to predct RP- HPLC retenton behavour. The successonal sectons wll gve nsght nto pseudo-lqud molecule development and nto the extent to whch predcton of chromatographc systems s possble Effect of pseudo-lqud molecule structure and composton on predcton qualty As the approach of statonary phase smulaton va pseudo-lqud molecules and successve COSMO-RS calculaton has never been performed before, there are bascally nether clues nor rules from lterature to be found of how to create pseudo-lqud molecules. Ths secton s therefore meant to present the way that has been pursued to fnd approprate pseudo-lqud molecules whch are ft to characterze the reversed phases under nvestgaton. Here, partcular attenton has been devoted onto nfluence of molecule chan length, the degree of molecular branchng, the utlsaton of polar groups and the possblty of usng a molecule set nstead of a sngle pseudo-lqud molecule. As has been shown n the recent secton, dodecanol s more sutable to represent a reversed phase chromatographc system wth ts polar nteractons than a non-polar dodecane molecule. Therefore, wthn upcomng calculatons, more preferably alcohols wll be used as pseudo-lqud model molecules.

105 89 Results & Dscusson Furthermore, usng a molecule made out of a C12 chan mght not be deal to represent a statonary phase that conssts of surface bound C18 alkyl chans as nteractng elements. Hence, Fg shows the nfluence of dfference n pseudo-lqud molecule chan length on the predcton qualty. Fgure 4.14: Correlaton coeffcent between log K COSMO-RS and log k exp values for a RP-HPLC system consstng of a statonary ODS phase (Hypersl C18) and a acetontrle-water moble phase at T = 25 C; Water content x water of the polar acetontrle-water phase s 0,75. Varable calculaton nput parameter s the pseudolqud molecule used to smulate the reversed statonary phase. Expermental log k data was taken from [Tan 1996] If compared to expermental values from a reversed phase system consstng of an aqueous moble phase and an ODS statonary phase, ncrease n pseudo-lqud molecule chan length results nto an ncrease n predcton qualty, as can be seen n Fg It s further shown that the amount of predcton qualty ncrease wll dmnsh wth growng alkyl chan length. Ths fndng can be explaned by the fact that one sde of the alkyl chan s bound to the surface of the slca partcles, partally nhbtng a lqud lke movement of the chan. In other words, wthn a bound C18 alkyl chan, not all C atoms wll have lqud lke freedom of movement. Therefore, the nteracton characterstca of a bound C18 chan could eventually be well resembled also by a shorter alkyl chan n lqud state. In Fg. 4.15, the pseudo-lqud molecule alkyl chan length s left constant, whle the degree of branchng s beng vared. Fg s therefore amed to depct the nfluence of branchng onto the qualty of predcton.

106 90 Results & Dscusson Fgure 4.15: Correlaton coeffcent between log K COSMO-RS and log k exp values for RP-HPLC systems consstng of dfferent statonary phase (see Tab. 3.2) and moble phases, respectvely; Temperature s set to T = 25 C for all systems; Water content x water wthn acetontrle-water phase s 0,74 whle x water = 0,36 wthn both methanolwater phases; Varable calculaton nput parameter s the pseudo-lqud molecule used to smulate the reversed statonary phase; Expermental log k data was taken from [Tan 1996] and [Tsukahara 1993] The qualty of the predcton, expressed n terms of the correlaton coeffcent, s dependent on branchng and poston of the hydroxyl group wthn the pseudo-molecule, as can be seen from Fg Correlatons between expermental log k data, usng a methanol-water mxture as moble phase and COSMO-RS calculated log K show lttle nfluence of the mentoned hydroxyl group postonng onto the qualty of predcton. Own measurements were conducted usng a YMC C18 column (see Tab. 3.2) under otherwse equal condtons. The obtaned R² shows the same trend but wth relatvely smaller R² for 2-Slanoloctane and 3-Slanoloctane. By contrast, f an acetontrle-water mxture s beng used as moble phase, correlaton between expermental and predcted values depends strongly on O-H postonng wthn the structure of the pseudo-lqud molecule. To smulate the statonary phase, t s also possble to use a combnaton of dfferent pseudolqud molecule structures nstead of a sngle one. The number of possble combnatons that can be thought of s rather vast and for means of smplcty, only a few combnatons were consdered and compared to expermental data. An example can be found n Fg Here, a combnaton of octane and trmethylslanol structures was used n equal concentraton to smulate the statonary phase. Qualty of correlaton was comparable to that of an

107 91 Results & Dscusson octyldmethylslanol structure. Due to the fact that the sgma profles of the equmolar molecule combnaton and the sngle molecule are close matches, ths fndng s not surprsng. In summary t can be sad that n average, best correlaton coeffcents along wth smple confguraton can be reached by usng a termnal hydroxyl group wthn a sngle molecule structure. As nfluence of pseudo-molecule chan length and structure was explaned n Fgures 4.14 and 4.15, the next fgure s meant to depct the effect of varyng pseudo-lqud molecule polarty onto R². Fgure 4.16: Correlaton coeffcent between log K COSMO-RS and log k exp values for RP-HPLC systems consstng of dfferent statonary phases and moble phases, respectvely; Temperature s set to T = 25 C for all systems; Water content x water wthn acetontrle-water phase s 0,74 whle x water = 0,36 wthn both methanol-water phases; Varable calculaton nput parameter s the pseudo-lqud molecule used to smulate the reversed statonary phase; Expermental log k data was taken from [Tan 1996] and [Tsukahara 1993] The nfluence of change n pseudo-lqud polarty onto expermental k value predcton qualty can be seen from Fg Except for the octyldmethylslanol molecule whch leads to relatve constant R² values for all three systems under nvestgaton, all other pseudo-lqud molecule structures examned show strongly dfferng correlaton coeffcents results. Wth ncreasng number of S-bound hydroxyl groups, the polar character of the real chromatographc system wll be overestmated by the COSMO-RS calculatons and therefore lead to a decrease n R². Best average correlaton to expermental data was reached, f the pseudo-lqud molecule merely holds a sngle termnal hydroxyl group.

108 92 Results & Dscusson A summarzaton of the fndngs from Fgures 4.14 to 4.16 wll lead to the concluson that n average, a pseudo-lqud molecule consstng of an alkyl chan length equal to chan lengths wthn the real statonary phase and a termnal slanol group to express the polar nteracton wthn the reversed phase system wll lead to the most promsng correlaton coeffcents. Fg shows a schematc sketch of such a pseudo-lqud molecule deducton from the real phase model. Fgure 4.17: Deducton of a pseudo-lqud molecule from statonary phase structural group;[rethnger 2011] Effect of pseudo-lqud molecule conformaton selecton on predcton qualty Ths secton s meant to present the way that has been pursued to fnd an approprate rule for pseudo-lqud molecule conformaton selecton as t has already been conducted for solute and solvent molecules (Secton 4.1). As mentoned n Secton 2.5, alkyl chans bound to a slca support can not be assumed to always exst n a lnear shape whle exhbtng an deal fur-lke surface character. Ths leads to the queston of how to represent the reversed phase alkyl chan shape n terms of pseudo-lqud molecule conformaton. As to the lterature, deducton of bonded chan orentaton and conformaton from system parameters has not jet been acheved. Hence, a rule must be establshed that () allows for comparablty of predcton results and () gves constantly good predcton results, relatve to a random conformaton selecton. Fgures 4.18 and 4.19 show R² values for dfferent octadecyldmethylslanol conformatons that were used to smulated expermental data acqured at dfferent system condtons.

109 93 Results & Dscusson Fgure 4.18: Correlaton coeffcent between log K COSMO-RS and log k exp values; Temperature s set to T = 25 C; Varable calculaton nput parameter s the pseudolqud molecule conformaton; Expermental log k data was taken from [Tsukahara 1993] Fgure 4.19: Correlaton coeffcent between log K COSMO-RS and log k exp values; Temperature s set to T = 25 C; Varable calculaton nput parameter s the pseudolqud molecule conformaton; Expermental log k data was taken from [Tsukahara 1993] Both fgures above show a consstent trend regardng R² values. Although correlaton coeffcents n Fg show a lower average, R² between 0,89 and 0,93 stll wtness a hgh